Theorem 3anbi3d 1331

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 1327	1 |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981

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 983

This theorem is referenced by:  ceqsex3v  2815  ceqsex4v  2816  ceqsex8v  2818  vtocl3gaf  2842  mob  2955  ordsoexmid  4610  tfr1onlemaccex  6434  tfrcllemaccex  6447  fseq1m1p1  10217  summodc  11694  fsum3  11698  divalglemnn  12229  divalglemeunn  12232  divalglemex  12233  divalglemeuneg  12234  mhmlem  13450  ring1  13821  lmodlema  14054  ivthreinc  15117  dvmptfsum  15197