Theorem 3anbi3d 1359

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 1355	1 |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009

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 1011

This theorem is referenced by:  ceqsex3v  2865  ceqsex4v  2866  ceqsex8v  2868  vtocl3gaf  2892  mob  3008  ordsoexmid  4704  tfr1onlemaccex  6609  tfrcllemaccex  6622  fseq1m1p1  10480  pfxsuff1eqwrdeq  11449  summodc  12128  fsum3  12132  divalglemnn  12663  divalglemeunn  12666  divalglemex  12667  divalglemeuneg  12668  mhmlem  13894  ring1  14337  lmodlema  14601  ivthreinc  15669  dvmptfsum  15749