Theorem mpbirand 445

Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
mpbirand.1	|- ( ph -> ch )
mpbirand.2	|- ( ph -> ( ps <-> ( ch /\ th ) ) )

Assertion

Ref	Expression
mpbirand	|- ( ph -> ( ps <-> th ) )

Proof of Theorem mpbirand

Step	Hyp	Ref	Expression
1		mpbirand.2	. 2 |- ( ph -> ( ps <-> ( ch /\ th ) ) )
2		mpbirand.1	. . 3 |- ( ph -> ch )
3	2	biantrurd 305	. 2 |- ( ph -> ( th <-> ( ch /\ th ) ) )
4	1, 3	bitr4d 191	1 |- ( ph -> ( ps <-> th ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  fvdifsuppst  6484  2omap  7318  pfxsuffeqwrdeq  11470  fprodssdc  12357  dvdsr2d  14402  ellspsn5b  14746  psrbagfi  15059  psrbaglecl  15060  psrbagcon  15062  txmetcn  15620