Theorem a2and 558

Description: Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)

Hypotheses

Ref	Expression
a2and.1	|- ( ph -> ( ( ps /\ rh ) -> ( ta -> th ) ) )
a2and.2	|- ( ph -> ( ( ps /\ rh ) -> ch ) )

Assertion

Ref	Expression
a2and	|- ( ph -> ( ( ( ps /\ ch ) -> ta ) -> ( ( ps /\ rh ) -> th ) ) )

Proof of Theorem a2and

Step	Hyp	Ref	Expression
1		a2and.2	. . . . . . 7 |- ( ph -> ( ( ps /\ rh ) -> ch ) )
2	1	expd 258	. . . . . 6 |- ( ph -> ( ps -> ( rh -> ch ) ) )
3	2	imdistand 447	. . . . 5 |- ( ph -> ( ( ps /\ rh ) -> ( ps /\ ch ) ) )
4	3	imp 124	. . . 4 |- ( ( ph /\ ( ps /\ rh ) ) -> ( ps /\ ch ) )
5		a2and.1	. . . . 5 |- ( ph -> ( ( ps /\ rh ) -> ( ta -> th ) ) )
6	5	imp 124	. . . 4 |- ( ( ph /\ ( ps /\ rh ) ) -> ( ta -> th ) )
7	4, 6	embantd 56	. . 3 |- ( ( ph /\ ( ps /\ rh ) ) -> ( ( ( ps /\ ch ) -> ta ) -> th ) )
8	7	ex 115	. 2 |- ( ph -> ( ( ps /\ rh ) -> ( ( ( ps /\ ch ) -> ta ) -> th ) ) )
9	8	com23 78	1 |- ( ph -> ( ( ( ps /\ ch ) -> ta ) -> ( ( ps /\ rh ) -> th ) ) )

Syntax hints:   -> wi 4   /\ wa 104

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: (None)