Theorem reueq1f 1785

Description: Equality theorem for restricted uniqueness 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
reueq1f	|- (A = B -> (E!x e. A ph <-> E!x e. B ph))

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

Proof of Theorem reueq1f

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 1565	. . 3 |- (A = B -> A.x A = B)
4		eleq2 1535	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	anbi1d 617	. . 3 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
6	3, 5	eubid 1385	. 2 |- (A = B -> (E!x(x e. A /\ ph) <-> E!x(x e. B /\ ph)))
7		df-reu 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
8		df-reu 1651	. 2 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
9	6, 7, 8	3bitr4g 555	1 |- (A = B -> (E!x e. A ph <-> E!x e. B ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E!weu 1380  E!wreu 1647

This theorem is referenced by:  reueq1 1788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-eu 1382  df-cleq 1469  df-clel 1472  df-reu 1651

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