Theorem rmoeq1 2689

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
rmoeq1	|- ( 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 rmoeq1

Step	Hyp	Ref	Expression
1		nfcv 2332	. 2 |- F/_ x A
2		nfcv 2332	. 2 |- F/_ x B
3	1, 2	rmoeq1f 2685	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E*wrmo 2471

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rmo 2476

This theorem is referenced by:  rmoeqd  2697