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Theorem eq2tri 2141
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 262 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( B  =  D  /\  A  =  C )
)
2 eq2tr.1 . . . 4  |-  ( A  =  C  ->  D  =  F )
32eqeq2d 2093 . . 3  |-  ( A  =  C  ->  ( B  =  D  <->  B  =  F ) )
43pm5.32i 442 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( A  =  C  /\  B  =  F )
)
5 eq2tr.2 . . . 4  |-  ( B  =  D  ->  C  =  G )
65eqeq2d 2093 . . 3  |-  ( B  =  D  ->  ( A  =  C  <->  A  =  G ) )
76pm5.32i 442 . 2  |-  ( ( B  =  D  /\  A  =  C )  <->  ( B  =  D  /\  A  =  G )
)
81, 4, 73bitr3i 208 1  |-  ( ( A  =  C  /\  B  =  F )  <->  ( B  =  D  /\  A  =  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285
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 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075
This theorem is referenced by:  xpassen  6374
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