Theorem rmoeq1 3141

Description: Equality theorem for restricted uniqueness 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 2764	. 2 |- F/_ x A
2		nfcv 2764	. 2 |- F/_ x B
3	1, 2	rmoeq1f 3137	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920

This theorem is referenced by:  rmoeqd  3149  poimirlem2  33411