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Theorem rmoeqd 2760
Description: Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
raleqd.1  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rmoeqd  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ps ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rmoeqd
StepHypRef Expression
1 rmoeq1 2752 . 2  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32rmobidv 2742 . 2  |-  ( A  =  B  ->  ( E* x  e.  B ph 
<->  E* x  e.  B ps ) )
41, 3bitrd 244 1  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   E*wrmo 2559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rmo 2564
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