Theorem reueqd 3148

Description: Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reueqd

Step	Hyp	Ref	Expression
1		reueq1 3140	. 2 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	reubidv 3126	. 2 |- ( A = B -> ( E! x e. B ph <-> E! x e. B ps ) )
4	1, 3	bitrd 268	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-cleq 2615  df-clel 2618  df-nfc 2753  df-reu 2919

This theorem is referenced by:  aceq1  8940