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 456	. . . . 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 413	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
7	6	a2d 29	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  seqcl2  13929  seqfveq2  13933  seqshft2  13937  monoord  13941  seqsplit  13944  seqid2  13957  seqhomo  13958  sylow1lem1  19512  imasdsf1olem  24289  ovolicc2lem3  25448  dvnres  25861  cvmliftlem7  35356  cvmliftlem10  35359  monoordxrv  45603  smonoord  47495