Theorem mp3anl2 1455

Description: An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
mp3anl2.1	⊢ 𝜓
mp3anl2.2	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
mp3anl2	⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem mp3anl2

Step	Hyp	Ref	Expression
1		mp3anl2.1	. . 3 ⊢ 𝜓
2		mp3anl2.2	. . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
3	2	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
4	1, 3	mp3an2 1448	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 → 𝜏))
5	4	imp 406	1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 396  df-3an 1088

This theorem is referenced by:  mp3anr2  1458  preleq  9617  1dvds  16221  bcs2  30869  nmopub2tALT  31596  nmfnleub2  31613  nmophmi  31718  nmopcoadji  31788  atordi  32071  mdsymlem5  32094