Theorem anim12ii 618

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		pm3.43 473	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜓 → 𝜏)) → (𝜓 → (𝜒 ∧ 𝜏)))
4	1, 2, 3	syl2an 596	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 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:  im2anan9  620  pm5.31r  832  2mo  2648  elex22  3506  disj  4450  tz7.2  5668  funcnvuni  7954  upgrwlkdvdelem  29756  funressnfv  47055