Theorem anim12ii 630

Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)

Hypotheses

Ref	Expression
anim12ii.1	⊢ (𝜑 → (𝜓 → 𝜒))
anim12ii.2	⊢ (𝜃 → (𝜓 → 𝜏))

Assertion

Ref	Expression
anim12ii	⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏)))

Proof of Theorem anim12ii

Step	Hyp	Ref	Expression
1		anim12ii.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		anim12ii.2	. 2 ⊢ (𝜃 → (𝜓 → 𝜏))
3		pm3.43 479	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜓 → 𝜏)) → (𝜓 → (𝜒 ∧ 𝜏)))
4	1, 2, 3	syl2an 608	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  im2anan9  632  pm5.31r  845  2mo  2678  elex22  3481  tz7.2  5646  funcnvuni  7931  elirrv  9562  upgrwlkdvdelem  30125  axprALT2  35537  funressnfv  47813