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Theorem ch2var 12988
Description: Implicit substitution of  y for  x and  t for  z into a theorem. (Contributed by BJ, 17-Oct-2019.)
Hypotheses
Ref Expression
ch2var.nfx  |-  F/ x ps
ch2var.nfz  |-  F/ z ps
ch2var.maj  |-  ( ( x  =  y  /\  z  =  t )  ->  ( ph  <->  ps )
)
ch2var.min  |-  ph
Assertion
Ref Expression
ch2var  |-  ps
Distinct variable groups:    x, z    x, t
Allowed substitution hints:    ph( x, y, z, t)    ps( x, y, z, t)

Proof of Theorem ch2var
StepHypRef Expression
1 ch2var.nfx . . 3  |-  F/ x ps
2 ch2var.nfz . . 3  |-  F/ z ps
3 ch2var.maj . . . 4  |-  ( ( x  =  y  /\  z  =  t )  ->  ( ph  <->  ps )
)
43biimpd 143 . . 3  |-  ( ( x  =  y  /\  z  =  t )  ->  ( ph  ->  ps ) )
51, 2, 42spim 12987 . 2  |-  ( A. z A. x ph  ->  ps )
6 ch2var.min . . 3  |-  ph
76ax-gen 1425 . 2  |-  A. x ph
85, 7mpg 1427 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329   F/wnf 1436
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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  ch2varv  12989
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