Theorem anim12ii 619

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		pm3.43 476	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜓 → 𝜏)) → (𝜓 → (𝜒 ∧ 𝜏)))
4	1, 2, 3	syl2an 597	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  im2anan9  621  pm5.31r  829  euimOLD  2701  2mo  2732  elex22  3520  tz7.2  5542  funcnvuni  7639  upgrwlkdvdelem  27520  funressnfv  43285