Theorem 3anbi12d 894

Description: Deduction conjoining and adding a conjunct to equivalences.

Hypotheses

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

Assertion

Ref	Expression
3anbi12d	|- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ et)))

Proof of Theorem 3anbi12d

Step	Hyp	Ref	Expression
1		3anbi12d.1	. 2 |- (ph -> (ps <-> ch))
2		3anbi12d.2	. 2 |- (ph -> (th <-> ta))
3		pm4.2d 171	. 2 |- (ph -> (et <-> et))
4	1, 2, 3	3anbi123d 893	1 |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ et)))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775

This theorem is referenced by:  3anbi1d 897  3anbi2d 898  caufval 7926  symgval 10403  cnvhmph 10527  isfil 10558  isded 10669  dedi 10670  ismonb2 10740  isepib2 10750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

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