Theorem 3anbi3d 1318

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 1314	1 |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978

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 980

This theorem is referenced by:  ceqsex3v  2779  ceqsex4v  2780  ceqsex8v  2782  vtocl3gaf  2806  mob  2919  ordsoexmid  4559  tfr1onlemaccex  6344  tfrcllemaccex  6357  fseq1m1p1  10088  summodc  11382  fsum3  11386  divalglemnn  11913  divalglemeunn  11916  divalglemex  11917  divalglemeuneg  11918  mhmlem  12906  ring1  13136