MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl3gf Structured version   Unicode version

Theorem vtocl3gf 3139
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gf.a  |-  F/_ x A
vtocl3gf.b  |-  F/_ y A
vtocl3gf.c  |-  F/_ z A
vtocl3gf.d  |-  F/_ y B
vtocl3gf.e  |-  F/_ z B
vtocl3gf.f  |-  F/_ z C
vtocl3gf.1  |-  F/ x ps
vtocl3gf.2  |-  F/ y ch
vtocl3gf.3  |-  F/ z th
vtocl3gf.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl3gf.5  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl3gf.6  |-  ( z  =  C  ->  ( ch 
<->  th ) )
vtocl3gf.7  |-  ph
Assertion
Ref Expression
vtocl3gf  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  th )

Proof of Theorem vtocl3gf
StepHypRef Expression
1 elex 3087 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 vtocl3gf.d . . . 4  |-  F/_ y B
3 vtocl3gf.e . . . 4  |-  F/_ z B
4 vtocl3gf.f . . . 4  |-  F/_ z C
5 vtocl3gf.b . . . . . 6  |-  F/_ y A
65nfel1 2632 . . . . 5  |-  F/ y  A  e.  _V
7 vtocl3gf.2 . . . . 5  |-  F/ y ch
86, 7nfim 1858 . . . 4  |-  F/ y ( A  e.  _V  ->  ch )
9 vtocl3gf.c . . . . . 6  |-  F/_ z A
109nfel1 2632 . . . . 5  |-  F/ z  A  e.  _V
11 vtocl3gf.3 . . . . 5  |-  F/ z th
1210, 11nfim 1858 . . . 4  |-  F/ z ( A  e.  _V  ->  th )
13 vtocl3gf.5 . . . . 5  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1413imbi2d 316 . . . 4  |-  ( y  =  B  ->  (
( A  e.  _V  ->  ps )  <->  ( A  e.  _V  ->  ch )
) )
15 vtocl3gf.6 . . . . 5  |-  ( z  =  C  ->  ( ch 
<->  th ) )
1615imbi2d 316 . . . 4  |-  ( z  =  C  ->  (
( A  e.  _V  ->  ch )  <->  ( A  e.  _V  ->  th )
) )
17 vtocl3gf.a . . . . 5  |-  F/_ x A
18 vtocl3gf.1 . . . . 5  |-  F/ x ps
19 vtocl3gf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
20 vtocl3gf.7 . . . . 5  |-  ph
2117, 18, 19, 20vtoclgf 3134 . . . 4  |-  ( A  e.  _V  ->  ps )
222, 3, 4, 8, 12, 14, 16, 21vtocl2gf 3138 . . 3  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( A  e.  _V  ->  th ) )
231, 22mpan9 469 . 2  |-  ( ( A  e.  V  /\  ( B  e.  W  /\  C  e.  X
) )  ->  th )
24233impb 1184 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078
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-11 1782  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-nfc 2604  df-v 3080
This theorem is referenced by:  vtocl3gaf  3145
  Copyright terms: Public domain W3C validator