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Theorem eq2tri 2448
Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
Hypotheses
Ref Expression
eq2tr.1  |-  ( A  =  C  ->  D  =  F )
eq2tr.2  |-  ( B  =  D  ->  C  =  G )
Assertion
Ref Expression
eq2tri  |-  ( ( A  =  C  /\  B  =  F )  <->  ( B  =  D  /\  A  =  G )
)

Proof of Theorem eq2tri
StepHypRef Expression
1 ancom 438 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( B  =  D  /\  A  =  C )
)
2 eq2tr.1 . . . 4  |-  ( A  =  C  ->  D  =  F )
32eqeq2d 2400 . . 3  |-  ( A  =  C  ->  ( B  =  D  <->  B  =  F ) )
43pm5.32i 619 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( A  =  C  /\  B  =  F )
)
5 eq2tr.2 . . . 4  |-  ( B  =  D  ->  C  =  G )
65eqeq2d 2400 . . 3  |-  ( B  =  D  ->  ( A  =  C  <->  A  =  G ) )
76pm5.32i 619 . 2  |-  ( ( B  =  D  /\  A  =  C )  <->  ( B  =  D  /\  A  =  G )
)
81, 4, 73bitr3i 267 1  |-  ( ( A  =  C  /\  B  =  F )  <->  ( B  =  D  /\  A  =  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649
This theorem is referenced by:  xpassen  7140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2382
  Copyright terms: Public domain W3C validator