Theorem pm5.32ri 490

Description: Distribution of implication over biconditional (inference rule).

Hypothesis

Ref	Expression
pm5.32i.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
pm5.32ri	|- ((ps /\ ph) <-> (ch /\ ph))

Proof of Theorem pm5.32ri

Step	Hyp	Ref	Expression
1		pm5.32i.1	. . 3 |- (ph -> (ps <-> ch))
2	1	pm5.32i 489	. 2 |- ((ph /\ ps) <-> (ph /\ ch))
3		ancom 333	. 2 |- ((ps /\ ph) <-> (ph /\ ps))
4		ancom 333	. 2 |- ((ch /\ ph) <-> (ph /\ ch))
5	2, 3, 4	3bitr4 158	1 |- ((ps /\ ph) <-> (ch /\ ph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  2eu5 1071  dfoprab2 3021  th3qlem1 3250  xpsnen 3339  pw2en 3348

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

This theorem depends on definitions:  df-bi 128  df-an 198

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