MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl3ga Structured version   Unicode version

Theorem vtocl3ga 3146
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 2616 . 2  |-  F/_ x A
2 nfcv 2616 . 2  |-  F/_ y A
3 nfcv 2616 . 2  |-  F/_ z A
4 nfcv 2616 . 2  |-  F/_ y B
5 nfcv 2616 . 2  |-  F/_ z B
6 nfcv 2616 . 2  |-  F/_ z C
7 nfv 1674 . 2  |-  F/ x ps
8 nfv 1674 . 2  |-  F/ y ch
9 nfv 1674 . 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 3145 1  |-  ( ( A  e.  D  /\  B  e.  R  /\  C  e.  S )  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758
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:  preq12bg  4162  prel12g  4163  pocl  4759  jensenlem2  22524
  Copyright terms: Public domain W3C validator