MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmoeq1f Unicode version

Theorem rmoeq1f 2735
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
rmoeq1f  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ph ) )

Proof of Theorem rmoeq1f
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2426 . . 3  |-  F/ x  A  =  B
4 eleq2 2344 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 685 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5mobid 2177 . 2  |-  ( A  =  B  ->  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  B  /\  ph )
) )
7 df-rmo 2551 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
8 df-rmo 2551 . 2  |-  ( E* x  e.  B ph  <->  E* x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 279 1  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E*wmo 2144   F/_wnfc 2406   E*wrmo 2546
This theorem is referenced by:  rmoeq1  2739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rmo 2551
  Copyright terms: Public domain W3C validator