Theorem csbcomg 2007

Description: Commutative law for double substitution into a class.

Assertion

Ref	Expression
csbcomg	|- ((A e. R /\ B e. S) -> [_A / x]_[_B / y]_C = [_B / y]_[_A / x]_C)

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

Proof of Theorem csbcomg

Step	Hyp	Ref	Expression
1		sbccomg 1979	. . . . 5 |- ((A e. V /\ B e. V) -> ([A / x][B / y]z e. C <-> [B / y][A / x]z e. C))
2		sbcel2g 2005	. . . . . . 7 |- (B e. V -> ([B / y]z e. C <-> z e. [_B / y]_C))
3	2	sbcbidv 1967	. . . . . 6 |- ((B e. V /\ A e. V) -> ([A / x][B / y]z e. C <-> [A / x]z e. [_B / y]_C))
4	3	ancoms 436	. . . . 5 |- ((A e. V /\ B e. V) -> ([A / x][B / y]z e. C <-> [A / x]z e. [_B / y]_C))
5		sbcel2g 2005	. . . . . 6 |- (A e. V -> ([A / x]z e. C <-> z e. [_A / x]_C))
6	5	sbcbidv 1967	. . . . 5 |- ((A e. V /\ B e. V) -> ([B / y][A / x]z e. C <-> [B / y]z e. [_A / x]_C))
7	1, 4, 6	3bitr3d 546	. . . 4 |- ((A e. V /\ B e. V) -> ([A / x]z e. [_B / y]_C <-> [B / y]z e. [_A / x]_C))
8		sbcel2g 2005	. . . . 5 |- (A e. V -> ([A / x]z e. [_B / y]_C <-> z e. [_A / x]_[_B / y]_C))
9	8	adantr 389	. . . 4 |- ((A e. V /\ B e. V) -> ([A / x]z e. [_B / y]_C <-> z e. [_A / x]_[_B / y]_C))
10		sbcel2g 2005	. . . . 5 |- (B e. V -> ([B / y]z e. [_A / x]_C <-> z e. [_B / y]_[_A / x]_C))
11	10	adantl 388	. . . 4 |- ((A e. V /\ B e. V) -> ([B / y]z e. [_A / x]_C <-> z e. [_B / y]_[_A / x]_C))
12	7, 9, 11	3bitr3d 546	. . 3 |- ((A e. V /\ B e. V) -> (z e. [_A / x]_[_B / y]_C <-> z e. [_B / y]_[_A / x]_C))
13	12	eqrdv 1466	. 2 |- ((A e. V /\ B e. V) -> [_A / x]_[_B / y]_C = [_B / y]_[_A / x]_C)
14		elisset 1808	. 2 |- (A e. R -> A e. V)
15		elisset 1808	. 2 |- (B e. S -> B e. V)
16	13, 14, 15	syl2an 454	1 |- ((A e. R /\ B e. S) -> [_A / x]_[_B / y]_C = [_B / y]_[_A / x]_C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  [wsbc 1166  Vcvv 1802  [_csb 1991

This theorem is referenced by:  csbfsumlem 6964

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-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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