Theorem reueq1 2732

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
reueq1	|- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem reueq1

Step	Hyp	Ref	Expression
1		nfcv 2374	. 2 |- F/_ x A
2		nfcv 2374	. 2 |- F/_ x B
3	1, 2	reueq1f 2728	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   E!wreu 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-cleq 2224  df-clel 2227  df-nfc 2363  df-reu 2517

This theorem is referenced by:  reueqd  2744  ringideu  14029  uspgredg2vlem  16070  uspgredg2v  16071