Theorem pm4.71d 393

Description: Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypothesis

Ref	Expression
pm4.71rd.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
pm4.71d	|- ( ph -> ( ps <-> ( ps /\ ch ) ) )

Proof of Theorem pm4.71d

Step	Hyp	Ref	Expression
1		pm4.71rd.1	. 2 |- ( ph -> ( ps -> ch ) )
2		pm4.71 389	. 2 |- ( ( ps -> ch ) <-> ( ps <-> ( ps /\ ch ) ) )
3	1, 2	sylib 122	1 |- ( ph -> ( ps <-> ( ps /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  difin2  3399  resopab2  4956  fcnvres  5401  resoprab2  5975  cndis  13881  cnpdis  13882  blpnf  14040