Theorem pm5.74i 443

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

Hypothesis

Ref	Expression
pm5.74i.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
pm5.74i	|- ((ph -> ps) <-> (ph -> ch))

Proof of Theorem pm5.74i

Step	Hyp	Ref	Expression
1		pm5.74i.1	. 2 |- (ph -> (ps <-> ch))
2		pm5.74 442	. 2 |- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))
3	1, 2	mpbi 164	1 |- ((ph -> ps) <-> (ph -> ch))

Syntax hints:   -> wi 2   <-> wb 127

This theorem is referenced by:  mpbidi 447  ibib 448  eqsal 833  sb6a 990  birala 1228  dfom2 2374  weinxp 2467  kmlem12 3591  kmlem13 3592  kmlem14 3593  uzind 4603

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

metamath.org