Theorem mp3anl2 1485

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 417	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
4	1, 3	mp3an2 1478	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 → 𝜏))
5	4	imp 411	1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103

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 401  df-3an 1105

This theorem is used by:  mp3anr2  1488  preleq  9581  1dvds  16332  bcs2  31543  nmopub2tALT  32270  nmfnleub2  32287  nmophmi  32392  nmopcoadji  32462  atordi  32745  mdsymlem5  32768