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Theorem subtr 30372
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 2627 . . . 4  |-  F/ x  A  =  B
4 subtr.3 . . . . 5  |-  F/_ x Y
5 subtr.4 . . . . 5  |-  F/_ x Z
64, 5nfeq 2627 . . . 4  |-  F/ x  Y  =  Z
73, 6nfim 1925 . . 3  |-  F/ x
( A  =  B  ->  Y  =  Z )
8 eqeq1 2458 . . . 4  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
9 subtr.5 . . . . 5  |-  ( x  =  A  ->  X  =  Y )
109eqeq1d 2456 . . . 4  |-  ( x  =  A  ->  ( X  =  Z  <->  Y  =  Z ) )
118, 10imbi12d 318 . . 3  |-  ( x  =  A  ->  (
( x  =  B  ->  X  =  Z )  <->  ( A  =  B  ->  Y  =  Z ) ) )
12 subtr.6 . . 3  |-  ( x  =  B  ->  X  =  Z )
131, 7, 11, 12vtoclgf 3162 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  Y  =  Z ) )
1413adantr 463 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   F/_wnfc 2602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108
This theorem is referenced by: (None)
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