Theorem 3anbi13d 1309

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 171	. 2 |- ( ph -> ( et <-> et ) )
3		3anbi12d.2	. 2 |- ( ph -> ( th <-> ta ) )
4	1, 2, 3	3anbi123d 1307	1 |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  3anbi3d  1313  tfr1onlemaccex  6327  tfrcllemaccex  6340  ltxrlt  7985  lmres  13042