Theorem sbcabel 1999

Description: Interchange class substitution and class abstraction.

Hypothesis

Ref	Expression
sbcabel.1	|- (z e. B -> A.x z e. B)

Assertion

Ref	Expression
sbcabel	|- (A e. C -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))

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

Proof of Theorem sbcabel

Step	Hyp	Ref	Expression
1		elisset 1820	. 2 |- (A e. C -> A e. V)
2		df-clel 1475	. . . . 5 |- ({y | ph} e. B <-> E.w(w = {y | ph} /\ w e. B))
3	2	sbcbii 1981	. . . 4 |- (A e. V -> ([A / x]{y | ph} e. B <-> [A / x]E.w(w = {y | ph} /\ w e. B)))
4		sbcexg 1978	. . . 4 |- (A e. V -> ([A / x]E.w(w = {y | ph} /\ w e. B) <-> E.w[A / x](w = {y | ph} /\ w e. B)))
5		sbcang 1974	. . . . . 6 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> ([A / x]w = {y | ph} /\ [A / x]w e. B)))
6		abeq2 1571	. . . . . . . . . 10 |- (w = {y | ph} <-> A.y(y e. w <-> ph))
7	6	sbcbii 1981	. . . . . . . . 9 |- (A e. V -> ([A / x]w = {y | ph} <-> [A / x]A.y(y e. w <-> ph)))
8		sbcalg 1977	. . . . . . . . 9 |- (A e. V -> ([A / x]A.y(y e. w <-> ph) <-> A.y[A / x](y e. w <-> ph)))
9		sbcbidig 1976	. . . . . . . . . . 11 |- (A e. V -> ([A / x](y e. w <-> ph) <-> ([A / x]y e. w <-> [A / x]ph)))
10		ax-17 973	. . . . . . . . . . . . 13 |- (y e. w -> A.x y e. w)
11	10	sbcgf 1989	. . . . . . . . . . . 12 |- (A e. V -> ([A / x]y e. w <-> y e. w))
12	11	bibi1d 621	. . . . . . . . . . 11 |- (A e. V -> (([A / x]y e. w <-> [A / x]ph) <-> (y e. w <-> [A / x]ph)))
13	9, 12	bitrd 530	. . . . . . . . . 10 |- (A e. V -> ([A / x](y e. w <-> ph) <-> (y e. w <-> [A / x]ph)))
14	13	albidv 1280	. . . . . . . . 9 |- (A e. V -> (A.y[A / x](y e. w <-> ph) <-> A.y(y e. w <-> [A / x]ph)))
15	7, 8, 14	3bitrd 546	. . . . . . . 8 |- (A e. V -> ([A / x]w = {y | ph} <-> A.y(y e. w <-> [A / x]ph)))
16		abeq2 1571	. . . . . . . 8 |- (w = {y | [A / x]ph} <-> A.y(y e. w <-> [A / x]ph))
17	15, 16	syl6bbr 540	. . . . . . 7 |- (A e. V -> ([A / x]w = {y | ph} <-> w = {y | [A / x]ph}))
18		ax-17 973	. . . . . . . . 9 |- (z e. w -> A.x z e. w)
19		sbcabel.1	. . . . . . . . 9 |- (z e. B -> A.x z e. B)
20	18, 19	hbel 1569	. . . . . . . 8 |- (w e. B -> A.x w e. B)
21	20	sbcgf 1989	. . . . . . 7 |- (A e. V -> ([A / x]w e. B <-> w e. B))
22	17, 21	anbi12d 630	. . . . . 6 |- (A e. V -> (([A / x]w = {y | ph} /\ [A / x]w e. B) <-> (w = {y | [A / x]ph} /\ w e. B)))
23	5, 22	bitrd 530	. . . . 5 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> (w = {y | [A / x]ph} /\ w e. B)))
24	23	exbidv 1281	. . . 4 |- (A e. V -> (E.w[A / x](w = {y | ph} /\ w e. B) <-> E.w(w = {y | [A / x]ph} /\ w e. B)))
25	3, 4, 24	3bitrd 546	. . 3 |- (A e. V -> ([A / x]{y | ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B)))
26		df-clel 1475	. . 3 |- ({y | [A / x]ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B))
27	25, 26	syl6bbr 540	. 2 |- (A e. V -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))
28	1, 27	syl 10	1 |- (A e. C -> ([A / x]{y | ph} e. B <-> {y | [A / x]ph} e. B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  {cab 1466  Vcvv 1814

This theorem is referenced by:  csbexg 2011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945

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