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Theorem 2spim 11024
Description: Double substitution, as in spim 1670. (Contributed by BJ, 17-Oct-2019.)
Hypotheses
Ref Expression
2spim.nfx  |-  F/ x ch
2spim.nfz  |-  F/ z ch
2spim.1  |-  ( ( x  =  y  /\  z  =  t )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
2spim  |-  ( A. z A. x ps  ->  ch )
Distinct variable groups:    x, z    x, t
Allowed substitution hints:    ps( x, y, z, t)    ch( x, y, z, t)

Proof of Theorem 2spim
StepHypRef Expression
1 2spim.nfz . 2  |-  F/ z ch
2 2spim.nfx . . . 4  |-  F/ x ch
32a1i 9 . . 3  |-  ( z  =  t  ->  F/ x ch )
4 2spim.1 . . . . 5  |-  ( ( x  =  y  /\  z  =  t )  ->  ( ps  ->  ch ) )
54expcom 114 . . . 4  |-  ( z  =  t  ->  (
x  =  y  -> 
( ps  ->  ch ) ) )
65alrimiv 1799 . . 3  |-  ( z  =  t  ->  A. x
( x  =  y  ->  ( ps  ->  ch ) ) )
73, 6spimd 11023 . 2  |-  ( z  =  t  ->  ( A. x ps  ->  ch ) )
81, 7spim 1670 1  |-  ( A. z A. x ps  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   F/wnf 1392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  ch2var  11025
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