Theorem animpimp2impd 549

Description: Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
animpimp2impd.1	|- ( ( ps /\ ph ) -> ( ch -> ( th -> et ) ) )
animpimp2impd.2	|- ( ( ps /\ ( ph /\ th ) ) -> ( et -> ta ) )

Assertion

Ref	Expression
animpimp2impd	|- ( ph -> ( ( ps -> ch ) -> ( ps -> ( th -> ta ) ) ) )

Proof of Theorem animpimp2impd

Step	Hyp	Ref	Expression
1		animpimp2impd.1	. . . 4 |- ( ( ps /\ ph ) -> ( ch -> ( th -> et ) ) )
2		animpimp2impd.2	. . . . . 6 |- ( ( ps /\ ( ph /\ th ) ) -> ( et -> ta ) )
3	2	expr 373	. . . . 5 |- ( ( ps /\ ph ) -> ( th -> ( et -> ta ) ) )
4	3	a2d 26	. . . 4 |- ( ( ps /\ ph ) -> ( ( th -> et ) -> ( th -> ta ) ) )
5	1, 4	syld 45	. . 3 |- ( ( ps /\ ph ) -> ( ch -> ( th -> ta ) ) )
6	5	expcom 115	. 2 |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
7	6	a2d 26	1 |- ( ph -> ( ( ps -> ch ) -> ( ps -> ( th -> ta ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103

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 is referenced by:  seq3fveq2  10404  seq3shft2  10408  seq3split  10414  seq3id2  10444