Theorem reueq1 2578

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 2235	. 2 |- F/_ x A
2		nfcv 2235	. 2 |- F/_ x B
3	1, 2	reueq1f 2574	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1296   E!wreu 2372

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-cleq 2088  df-clel 2091  df-nfc 2224  df-reu 2377

This theorem is referenced by:  reueqd  2586