Theorem im2anan9r 621

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)

Hypotheses

Ref	Expression
im2an9.1	⊢ (𝜑 → (𝜓 → 𝜒))
im2an9.2	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
im2anan9r	⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Proof of Theorem im2anan9r

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		im2an9.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
3	1, 2	im2anan9 620	. 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))
4	3	ancoms 459	1 ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  pssnn  8951  pssnnOLD  9040  lbreu  11925  catideu  17384  exidu1  36014  rngoideu  36061