Theorem animpimp2impd 846

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 460	. . . . 5 ⊢ ((𝜓 ∧ 𝜑) → (𝜃 → (𝜂 → 𝜏)))
4	3	a2d 29	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → ((𝜃 → 𝜂) → (𝜃 → 𝜏)))
5	1, 4	syld 47	. . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜏)))
6	5	expcom 417	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
7	6	a2d 29	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))

Syntax hints:   → wi 4   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  seqcl2  13594  seqfveq2  13598  seqshft2  13602  monoord  13606  seqsplit  13609  seqid2  13622  seqhomo  13623  sylow1lem1  18987  imasdsf1olem  23271  ovolicc2lem3  24416  dvnres  24828  cvmliftlem7  32966  cvmliftlem10  32969  monoordxrv  42697  smonoord  44496