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Theorem vtocl3ga 2730
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl3ga.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl3ga.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl3ga.3  |-  ( z  =  C  ->  ( ch 
<->  th ) )
vtocl3ga.4  |-  ( ( x  e.  D  /\  y  e.  R  /\  z  e.  S )  ->  ph )
Assertion
Ref Expression
vtocl3ga  |-  ( ( A  e.  D  /\  B  e.  R  /\  C  e.  S )  ->  th )
Distinct variable groups:    x, y, z, A    y, B, z   
z, C    x, D, y, z    x, R, y, z    x, S, y, z    ps, x    ch, y    th, z
Allowed substitution hints:    ph( x, y, z)    ps( y, z)    ch( x, z)    th( x, y)    B( x)    C( x, y)

Proof of Theorem vtocl3ga
StepHypRef Expression
1 nfcv 2258 . 2  |-  F/_ x A
2 nfcv 2258 . 2  |-  F/_ y A
3 nfcv 2258 . 2  |-  F/_ z A
4 nfcv 2258 . 2  |-  F/_ y B
5 nfcv 2258 . 2  |-  F/_ z B
6 nfcv 2258 . 2  |-  F/_ z C
7 nfv 1493 . 2  |-  F/ x ps
8 nfv 1493 . 2  |-  F/ y ch
9 nfv 1493 . 2  |-  F/ z th
10 vtocl3ga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
11 vtocl3ga.2 . 2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
12 vtocl3ga.3 . 2  |-  ( z  =  C  ->  ( ch 
<->  th ) )
13 vtocl3ga.4 . 2  |-  ( ( x  e.  D  /\  y  e.  R  /\  z  e.  S )  ->  ph )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13vtocl3gaf 2729 1  |-  ( ( A  e.  D  /\  B  e.  R  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  preq12bg  3670  pocl  4195  sowlin  4212
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