Theorem 3anbi1d 1256

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

Hypothesis

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

Assertion

Ref	Expression
3anbi1d	|- (ph -> ((ps /\ th /\ ta) <-> (ch /\ th /\ ta)))

Proof of Theorem 3anbi1d

Step	Hyp	Ref	Expression
1		3anbi1d.1	. 2 |- (ph -> (ps <-> ch))
2		biidd 228	. 2 |- (ph -> (th <-> th))
3	1, 2	3anbi12d 1253	1 |- (ph -> ((ps /\ th /\ ta) <-> (ch /\ th /\ ta)))

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by:  vtocl3gaf  2924  opkelins2kg  4252  opkelins3kg  4253  opkelsikg  4265  sikss1c1c  4268  brsi  4762  funsi  5521  brsnsi  5774  fnpprod  5844  ovce  6173