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

Theorem reueq1f 2971
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 2623 . . 3  |-  F/ x  A  =  B
4 eleq2 2538 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 719 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5eubid 2337 . 2  |-  ( A  =  B  ->  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  B  /\  ph )
) )
7 df-reu 2763 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
8 df-reu 2763 . 2  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 296 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E!weu 2319   F/_wnfc 2599   E!wreu 2758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-eu 2323  df-cleq 2464  df-clel 2467  df-nfc 2601  df-reu 2763
This theorem is referenced by:  reueq1  2975
  Copyright terms: Public domain W3C validator