Theorem rexeq1f 1781

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
raleq1f.1	|- (y e. A -> A.x y e. A)
raleq1f.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
rexeq1f	|- (A = B -> (E.x e. A ph <-> E.x e. B ph))

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

Proof of Theorem rexeq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- (y e. A -> A.x y e. A)
2		raleq1f.2	. . . 4 |- (y e. B -> A.x y e. B)
3	1, 2	hbeq 1562	. . 3 |- (A = B -> A.x A = B)
4		eleq2 1532	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	anbi1d 616	. . 3 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
6	3, 5	exbid 1103	. 2 |- (A = B -> (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ph)))
7		df-rex 1647	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
8		df-rex 1647	. 2 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
9	6, 7, 8	3bitr4g 554	1 |- (A = B -> (E.x e. A ph <-> E.x e. B ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  rexeq1 1784  zfrep6 3606

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-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470  df-rex 1647

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