Theorem reueq1 2745

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 2386	. 2 |- F/_ x A
2		nfcv 2386	. 2 |- F/_ x B
3	1, 2	reueq1f 2741	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E!wreu 2524

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-cleq 2227  df-clel 2230  df-nfc 2375  df-reu 2529

This theorem is referenced by:  reueqd  2757  ringideu  14182  uspgredg2vlem  16264  uspgredg2v  16265