Theorem reueq1 2751

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 2392	. 2 |- F/_ x A
2		nfcv 2392	. 2 |- F/_ x B
3	1, 2	reueq1f 2747	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   E!wreu 2530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-cleq 2231  df-clel 2234  df-nfc 2381  df-reu 2535

This theorem is used by:  reueqd  2763  ringideu  14321  uspgredg2vlem  16461  uspgredg2v  16462