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Theorem subtr2 30756
Description: Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
subtr.1  |-  F/_ x A
subtr.2  |-  F/_ x B
subtr2.3  |-  F/ x ps
subtr2.4  |-  F/ x ch
subtr2.5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
subtr2.6  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
subtr2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  ( ps  <->  ch )
) )

Proof of Theorem subtr2
StepHypRef Expression
1 subtr.1 . . 3  |-  F/_ x A
2 subtr.2 . . . . 5  |-  F/_ x B
31, 2nfeq 2602 . . . 4  |-  F/ x  A  =  B
4 subtr2.3 . . . . 5  |-  F/ x ps
5 subtr2.4 . . . . 5  |-  F/ x ch
64, 5nfbi 1992 . . . 4  |-  F/ x
( ps  <->  ch )
73, 6nfim 1978 . . 3  |-  F/ x
( A  =  B  ->  ( ps  <->  ch )
)
8 eqeq1 2433 . . . 4  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
9 subtr2.5 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
109bibi1d 320 . . . 4  |-  ( x  =  A  ->  (
( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
118, 10imbi12d 321 . . 3  |-  ( x  =  A  ->  (
( x  =  B  ->  ( ph  <->  ch )
)  <->  ( A  =  B  ->  ( ps  <->  ch ) ) ) )
12 subtr2.6 . . 3  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
131, 7, 11, 12vtoclgf 3143 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  ( ps  <->  ch ) ) )
1413adantr 466 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  ( ps  <->  ch )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   F/wnf 1663    e. wcel 1870   F/_wnfc 2577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089
This theorem is referenced by: (None)
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