Theorem im2anan9 562

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
im2anan9	⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Proof of Theorem im2anan9

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
3		im2an9.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	adantl 271	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜂))
5	2, 4	anim12d 328	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  im2anan9r  563  trin  3885  xpss12  4463  f1oun  5166  poxp  5873  brecop  6219  enq0sym  6622  genpdisj  6713