Theorem rmoeq1 2686

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 2329	. 2 |- F/_ x A
2		nfcv 2329	. 2 |- F/_ x B
3	1, 2	rmoeq1f 2682	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   E*wrmo 2468

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rmo 2473

This theorem is referenced by:  rmoeqd  2694