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

Theorem sbciegf 3324
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
sbciegf.1  |-  F/ x ps
sbciegf.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbciegf  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem sbciegf
StepHypRef Expression
1 sbciegf.1 . 2  |-  F/ x ps
2 sbciegf.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1592 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 sbciegft 3323 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
51, 3, 4mp3an23 1307 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   F/wnf 1590    e. wcel 1758   [.wsbc 3292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3078  df-sbc 3293
This theorem is referenced by:  sbcieg  3325  opelopabf  4720  eqerlem  7242  sbccomieg  29278  aomclem6  29559  fveqsb  29856  opelopabgf  30283  bnj919  32077  bnj1464  32154  bnj1123  32294  bnj1373  32338
  Copyright terms: Public domain W3C validator