Theorem pm5.32d 446

Description: Distribution of implication over biconditional (deduction form). (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		biimp 117	. . . 4 |- ( ( ch <-> th ) -> ( ch -> th ) )
3	1, 2	syl6 33	. . 3 |- ( ph -> ( ps -> ( ch -> th ) ) )
4	3	imdistand 444	. 2 |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) )
5		biimpr 129	. . . 4 |- ( ( ch <-> th ) -> ( th -> ch ) )
6	1, 5	syl6 33	. . 3 |- ( ph -> ( ps -> ( th -> ch ) ) )
7	6	imdistand 444	. 2 |- ( ph -> ( ( ps /\ th ) -> ( ps /\ ch ) ) )
8	4, 7	impbid 128	1 |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.32rd  447  pm5.32da  448  pm5.32  449  anbi2d  460  cbvex2  1902  cores  5089  isoini  5768  mpoeq123  5880  genpassl  7444  genpassu  7445  fzind  9279  btwnz  9283  elfzm11  9993  isprm2  11993  isprm3  11994