Theorem 3anbi3d 1352

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1d.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
3anbi3d	|- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )

Proof of Theorem 3anbi3d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 |- ( ph -> ( th <-> th ) )
2		3anbi1d.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	3anbi13d 1348	1 |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1004

This theorem is referenced by:  ceqsex3v  2843  ceqsex4v  2844  ceqsex8v  2846  vtocl3gaf  2870  mob  2985  ordsoexmid  4654  tfr1onlemaccex  6500  tfrcllemaccex  6513  fseq1m1p1  10303  pfxsuff1eqwrdeq  11247  summodc  11910  fsum3  11914  divalglemnn  12445  divalglemeunn  12448  divalglemex  12449  divalglemeuneg  12450  mhmlem  13667  ring1  14038  lmodlema  14272  ivthreinc  15335  dvmptfsum  15415