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Theorem subtr 26327
Description: Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
subtr.1  |-  F/_ x A
subtr.2  |-  F/_ x B
subtr.3  |-  F/_ x Y
subtr.4  |-  F/_ x Z
subtr.5  |-  ( x  =  A  ->  X  =  Y )
subtr.6  |-  ( x  =  B  ->  X  =  Z )
Assertion
Ref Expression
subtr  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )

Proof of Theorem subtr
StepHypRef Expression
1 subtr.1 . . 3  |-  F/_ x A
2 subtr.2 . . . . 5  |-  F/_ x B
31, 2nfeq 2439 . . . 4  |-  F/ x  A  =  B
4 subtr.3 . . . . 5  |-  F/_ x Y
5 subtr.4 . . . . 5  |-  F/_ x Z
64, 5nfeq 2439 . . . 4  |-  F/ x  Y  =  Z
73, 6nfim 1781 . . 3  |-  F/ x
( A  =  B  ->  Y  =  Z )
8 eqeq1 2302 . . . 4  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
9 subtr.5 . . . . 5  |-  ( x  =  A  ->  X  =  Y )
109eqeq1d 2304 . . . 4  |-  ( x  =  A  ->  ( X  =  Z  <->  Y  =  Z ) )
118, 10imbi12d 311 . . 3  |-  ( x  =  A  ->  (
( x  =  B  ->  X  =  Z )  <->  ( A  =  B  ->  Y  =  Z ) ) )
12 subtr.6 . . 3  |-  ( x  =  B  ->  X  =  Z )
131, 7, 11, 12vtoclgf 2855 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  Y  =  Z ) )
1413adantr 451 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   F/_wnfc 2419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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