Theorem sbccomg 1979

Description: Commutative law for double class substitution. (The proof was shortened by Eric Schmidt, 17-Jan-2007.)

Assertion

Ref	Expression
sbccomg	|- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Distinct variable groups:   y,A   x,B   x,y

Proof of Theorem sbccomg

Step	Hyp	Ref	Expression
1		sbccog 1942	. . . 4 |- (A e. V -> ([A / z][z / x][B / w][w / y]ph <-> [A / x][B / w][w / y]ph))
2	1	adantr 389	. . 3 |- ((A e. V /\ B e. V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / x][B / w][w / y]ph))
3		visset 1804	. . . . . . . . 9 |- z e. V
4		sbccomglem 1978	. . . . . . . . 9 |- ((B e. V /\ z e. V) -> ([B / w][z / x][w / y]ph <-> [z / x][B / w][w / y]ph))
5	3, 4	mpan2 694	. . . . . . . 8 |- (B e. V -> ([B / w][z / x][w / y]ph <-> [z / x][B / w][w / y]ph))
6		visset 1804	. . . . . . . . . 10 |- w e. V
7		sbccomglem 1978	. . . . . . . . . 10 |- ((z e. V /\ w e. V) -> ([z / x][w / y]ph <-> [w / y][z / x]ph))
8	3, 6, 7	mp2an 695	. . . . . . . . 9 |- ([z / x][w / y]ph <-> [w / y][z / x]ph)
9	8	sbcbii 1968	. . . . . . . 8 |- (B e. V -> ([B / w][z / x][w / y]ph <-> [B / w][w / y][z / x]ph))
10	5, 9	bitr3d 528	. . . . . . 7 |- (B e. V -> ([z / x][B / w][w / y]ph <-> [B / w][w / y][z / x]ph))
11	10	sbcbidv 1967	. . . . . 6 |- ((B e. V /\ A e. V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / z][B / w][w / y][z / x]ph))
12	11	ancoms 436	. . . . 5 |- ((A e. V /\ B e. V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / z][B / w][w / y][z / x]ph))
13		sbccomglem 1978	. . . . 5 |- ((A e. V /\ B e. V) -> ([A / z][B / w][w / y][z / x]ph <-> [B / w][A / z][w / y][z / x]ph))
14		sbccomglem 1978	. . . . . . 7 |- ((A e. V /\ w e. V) -> ([A / z][w / y][z / x]ph <-> [w / y][A / z][z / x]ph))
15	6, 14	mpan2 694	. . . . . 6 |- (A e. V -> ([A / z][w / y][z / x]ph <-> [w / y][A / z][z / x]ph))
16	15	sbcbidv 1967	. . . . 5 |- ((A e. V /\ B e. V) -> ([B / w][A / z][w / y][z / x]ph <-> [B / w][w / y][A / z][z / x]ph))
17	12, 13, 16	3bitrd 542	. . . 4 |- ((A e. V /\ B e. V) -> ([A / z][z / x][B / w][w / y]ph <-> [B / w][w / y][A / z][z / x]ph))
18		sbccog 1942	. . . . 5 |- (B e. V -> ([B / w][w / y][A / z][z / x]ph <-> [B / y][A / z][z / x]ph))
19	18	adantl 388	. . . 4 |- ((A e. V /\ B e. V) -> ([B / w][w / y][A / z][z / x]ph <-> [B / y][A / z][z / x]ph))
20		sbccog 1942	. . . . 5 |- (A e. V -> ([A / z][z / x]ph <-> [A / x]ph))
21	20	sbcbidv 1967	. . . 4 |- ((A e. V /\ B e. V) -> ([B / y][A / z][z / x]ph <-> [B / y][A / x]ph))
22	17, 19, 21	3bitrd 542	. . 3 |- ((A e. V /\ B e. V) -> ([A / z][z / x][B / w][w / y]ph <-> [B / y][A / x]ph))
23		sbccog 1942	. . . . 5 |- (B e. V -> ([B / w][w / y]ph <-> [B / y]ph))
24	23	sbcbidv 1967	. . . 4 |- ((B e. V /\ A e. V) -> ([A / x][B / w][w / y]ph <-> [A / x][B / y]ph))
25	24	ancoms 436	. . 3 |- ((A e. V /\ B e. V) -> ([A / x][B / w][w / y]ph <-> [A / x][B / y]ph))
26	2, 22, 25	3bitr3rd 547	. 2 |- ((A e. V /\ B e. V) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))
27		elisset 1808	. 2 |- (A e. C -> A e. V)
28		elisset 1808	. 2 |- (B e. D -> B e. V)
29	26, 27, 28	syl2an 454	1 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  [wsbc 1166  Vcvv 1802

This theorem is referenced by:  csbcomg 2007  csbabg 2033

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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