Theorem pm5.32d 437

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)

Hypothesis

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

Assertion

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

Proof of Theorem pm5.32d

Step	Hyp	Ref	Expression
1		pm5.32d.1	. . . 4 |- ( ph -> ( ps -> ( ch <-> th ) ) )
2		bi1 116	. . . 4 |- ( ( ch <-> th ) -> ( ch -> th ) )
3	1, 2	syl6 33	. . 3 |- ( ph -> ( ps -> ( ch -> th ) ) )
4	3	imdistand 435	. 2 |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) )
5		bi2 128	. . . 4 |- ( ( ch <-> th ) -> ( th -> ch ) )
6	1, 5	syl6 33	. . 3 |- ( ph -> ( ps -> ( th -> ch ) ) )
7	6	imdistand 435	. 2 |- ( ph -> ( ( ps /\ th ) -> ( ps /\ ch ) ) )
8	4, 7	impbid 127	1 |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.32rd  438  pm5.32da  439  pm5.32  440  anbi2d  451  cbvex2  1838  cores  4844  isoini  5477  mpt2eq123  5584  genpassl  6714  genpassu  6715  fzind  8462  btwnz  8466  elfzm11  9108  isprm2  10499  isprm3  10500