Theorem animpimp2impd 565

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

Hypotheses

Ref	Expression
animpimp2impd.1	⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂)))
animpimp2impd.2	⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏))

Assertion

Ref	Expression
animpimp2impd	⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))

Proof of Theorem animpimp2impd

Step	Hyp	Ref	Expression
1		animpimp2impd.1	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂)))
2		animpimp2impd.2	. . . . . 6 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏))
3	2	expr 375	. . . . 5 ⊢ ((𝜓 ∧ 𝜑) → (𝜃 → (𝜂 → 𝜏)))
4	3	a2d 26	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → ((𝜃 → 𝜂) → (𝜃 → 𝜏)))
5	1, 4	syld 45	. . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜏)))
6	5	expcom 116	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
7	6	a2d 26	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  seq3fveq2  10927  seqfveq2g  10929  seq3shft2  10933  seqshft2g  10934  seq3split  10940  seqsplitg  10941  seq3id2  10978  seqhomog  10982