Theorem 3anbi13d 1246

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)

Hypotheses

Ref	Expression
3anbi12d.1	|- ( ph -> ( ps <-> ch ) )
3anbi12d.2	|- ( ph -> ( th <-> ta ) )

Assertion

Ref	Expression
3anbi13d	|- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) )

Proof of Theorem 3anbi13d

Step	Hyp	Ref	Expression
1		3anbi12d.1	. 2 |- ( ph -> ( ps <-> ch ) )
2		biidd 170	. 2 |- ( ph -> ( et <-> et ) )
3		3anbi12d.2	. 2 |- ( ph -> ( th <-> ta ) )
4	1, 2, 3	3anbi123d 1244	1 |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 922

This theorem is referenced by:  3anbi3d  1250  tfr1onlemaccex  5997  tfrcllemaccex  6010  ltxrlt  7245