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Theorem 3anbi12d 1291
Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
Hypotheses
Ref Expression
3anbi12d.1  |-  ( ph  ->  ( ps  <->  ch )
)
3anbi12d.2  |-  ( ph  ->  ( th  <->  ta )
)
Assertion
Ref Expression
3anbi12d  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  <->  ( ch  /\ 
ta  /\  et )
) )

Proof of Theorem 3anbi12d
StepHypRef Expression
1 3anbi12d.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
2 3anbi12d.2 . 2  |-  ( ph  ->  ( th  <->  ta )
)
3 biidd 171 . 2  |-  ( ph  ->  ( et  <->  et )
)
41, 2, 33anbi123d 1290 1  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  <->  ( ch  /\ 
ta  /\  et )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 964
This theorem is referenced by:  3anbi1d  1294  3anbi2d  1295  fseq1m1p1  9868
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