Theorem reueqd 2636

Description: Equality deduction for restricted unique existential 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 2628	. 2 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	reubidv 2614	. 2 |- ( A = B -> ( E! x e. B ph <-> E! x e. B ps ) )
4	1, 3	bitrd 187	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E!wreu 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-cleq 2132  df-clel 2135  df-nfc 2270  df-reu 2423

This theorem is referenced by:  acexmidlemcase  5769