Theorem rmoeq1 2811

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 2490	. 2 |- F/_xA
2		nfcv 2490	. 2 |- F/_xB
3	1, 2	rmoeq1f 2807	1 |- (A = B -> (E*x e. A ph <-> E*x e. B ph))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rmo 2623

This theorem is referenced by:  rmoeqd  2819