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Theorem vtocl3ga 2640
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 2194 . 2  |-  F/_ x A
2 nfcv 2194 . 2  |-  F/_ y A
3 nfcv 2194 . 2  |-  F/_ z A
4 nfcv 2194 . 2  |-  F/_ y B
5 nfcv 2194 . 2  |-  F/_ z B
6 nfcv 2194 . 2  |-  F/_ z C
7 nfv 1437 . 2  |-  F/ x ps
8 nfv 1437 . 2  |-  F/ y ch
9 nfv 1437 . 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 2639 1  |-  ( ( A  e.  D  /\  B  e.  R  /\  C  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  preq12bg  3572  pocl  4068  sowlin  4085
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