Theorem animpimp2impd

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

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

Proof

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