Theorem sbcabel 1967

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 1792	. 2 |- (A e. C -> A e. V)
2		df-clel 1449	. . . . 5 |- ({y | ph} e. B <-> E.w(w = {y | ph} /\ w e. B))
3	2	sbcbii 1949	. . . 4 |- (A e. V -> ([A / x]{y | ph} e. B <-> [A / x]E.w(w = {y | ph} /\ w e. B)))
4		sbcexg 1946	. . . 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 1942	. . . . . 6 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> ([A / x]w = {y | ph} /\ [A / x]w e. B)))
6		abeq2 1544	. . . . . . . . . 10 |- (w = {y | ph} <-> A.y(y e. w <-> ph))
7	6	sbcbii 1949	. . . . . . . . 9 |- (A e. V -> ([A / x]w = {y | ph} <-> [A / x]A.y(y e. w <-> ph)))
8		sbcalg 1945	. . . . . . . . 9 |- (A e. V -> ([A / x]A.y(y e. w <-> ph) <-> A.y[A / x](y e. w <-> ph)))
9		sbcbidig 1944	. . . . . . . . . . 11 |- (A e. V -> ([A / x](y e. w <-> ph) <-> ([A / x]y e. w <-> [A / x]ph)))
10		ax-17 1190	. . . . . . . . . . . . 13 |- (y e. w -> A.x y e. w)
11	10	sbcgf 1957	. . . . . . . . . . . 12 |- (A e. V -> ([A / x]y e. w <-> y e. w))
12	11	bibi1d 617	. . . . . . . . . . 11 |- (A e. V -> (([A / x]y e. w <-> [A / x]ph) <-> (y e. w <-> [A / x]ph)))
13	9, 12	bitrd 526	. . . . . . . . . 10 |- (A e. V -> ([A / x](y e. w <-> ph) <-> (y e. w <-> [A / x]ph)))
14	13	albidv 1260	. . . . . . . . 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 542	. . . . . . . 8 |- (A e. V -> ([A / x]w = {y | ph} <-> A.y(y e. w <-> [A / x]ph)))
16		abeq2 1544	. . . . . . . 8 |- (w = {y | [A / x]ph} <-> A.y(y e. w <-> [A / x]ph))
17	15, 16	syl6bbr 536	. . . . . . 7 |- (A e. V -> ([A / x]w = {y | ph} <-> w = {y | [A / x]ph}))
18		ax-17 1190	. . . . . . . . 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 1542	. . . . . . . 8 |- (w e. B -> A.x w e. B)
21	20	sbcgf 1957	. . . . . . 7 |- (A e. V -> ([A / x]w e. B <-> w e. B))
22	17, 21	anbi12d 626	. . . . . 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 526	. . . . 5 |- (A e. V -> ([A / x](w = {y | ph} /\ w e. B) <-> (w = {y | [A / x]ph} /\ w e. B)))
24	23	exbidv 1261	. . . 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 542	. . 3 |- (A e. V -> ([A / x]{y | ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B)))
26		df-clel 1449	. . 3 |- ({y | [A / x]ph} e. B <-> E.w(w = {y | [A / x]ph} /\ w e. B))
27	25, 26	syl6bbr 536	. 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 950  E.wex 956   = wceq 1099   e. wcel 1105  [wsbc 1153  {cab 1440  Vcvv 1786

This theorem is referenced by:  csbexg 1979

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913

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