Theorem sbc5g 1950

Description: An equivalence for class substitution.

Assertion

Ref	Expression
sbc5g	|- (A e. B -> ([A / x]ph <-> E.x(x = A /\ ph)))

Distinct variable group:   x,A

Proof of Theorem sbc5g

Step	Hyp	Ref	Expression
1		biimt 730	. . . . . 6 |- (A e. V -> (ph <-> (A e. V -> ph)))
2	1	anbi2d 615	. . . . 5 |- (A e. V -> ((x = A /\ ph) <-> (x = A /\ (A e. V -> ph))))
3	2	exbidv 1277	. . . 4 |- (A e. V -> (E.x(x = A /\ ph) <-> E.x(x = A /\ (A e. V -> ph))))
4		biimt 730	. . . 4 |- (A e. V -> (E.x(x = A /\ ph) <-> (A e. V -> E.x(x = A /\ ph))))
5		ax-17 969	. . . . . 6 |- (y e. A -> A.x y e. A)
6	5	hbsbc1 1945	. . . . 5 |- ((A e. V -> [A / x]ph) -> A.x(A e. V -> [A / x]ph))
7		sbceq1a 1940	. . . . . 6 |- (x = A -> (ph <-> [A / x]ph))
8	7	imbi2d 611	. . . . 5 |- (x = A -> ((A e. V -> ph) <-> (A e. V -> [A / x]ph)))
9	6, 8	ceqsexg 1883	. . . 4 |- (A e. V -> (E.x(x = A /\ (A e. V -> ph)) <-> (A e. V -> [A / x]ph)))
10	3, 4, 9	3bitr3rd 548	. . 3 |- (A e. V -> ((A e. V -> [A / x]ph) <-> (A e. V -> E.x(x = A /\ ph))))
11	10	pm5.74rd 587	. 2 |- (A e. V -> (A e. V -> ([A / x]ph <-> E.x(x = A /\ ph))))
12		elisset 1813	. 2 |- (A e. B -> A e. V)
13	11, 12, 12	sylc 68	1 |- (A e. B -> ([A / x]ph <-> E.x(x = A /\ ph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168  Vcvv 1807

This theorem is referenced by:  sbc5 1952  sbc2or 1954  sbciegft 1955  sbcgf 1982  sbccomglem 1984

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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