MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reueq1f Structured version   Unicode version

Theorem reueq1f 2920
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
reueq1f  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )

Proof of Theorem reueq1f
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2591 . . 3  |-  F/ x  A  =  B
4 eleq2 2504 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 704 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5eubid 2274 . 2  |-  ( A  =  B  ->  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  B  /\  ph )
) )
7 df-reu 2727 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
8 df-reu 2727 . 2  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 288 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!weu 2253   F/_wnfc 2571   E!wreu 2722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-cleq 2436  df-clel 2439  df-nfc 2573  df-reu 2727
This theorem is referenced by:  reueq1  2924
  Copyright terms: Public domain W3C validator