Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ceqsalgALT Structured version   Unicode version

Theorem bj-ceqsalgALT 34802
Description: Alternate proof of bj-ceqsalg 34801. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalg.1  |-  F/ x ps
bj-ceqsalg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bj-ceqsalgALT  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem bj-ceqsalgALT
StepHypRef Expression
1 bj-ceqsalg.1 . 2  |-  F/ x ps
2 bj-ceqsalg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1626 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 bj-ceqsalt 34798 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1312 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1397    = wceq 1399   F/wnf 1624    e. wcel 1826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-12 1862
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-clel 2377
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator