Theorem mp3anl1 1456

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

Hypotheses

Ref	Expression
mp3anl1.1	⊢ 𝜑
mp3anl1.2	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
mp3anl1	⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem mp3anl1

Step	Hyp	Ref	Expression
1		mp3anl1.1	. . 3 ⊢ 𝜑
2		mp3anl1.2	. . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
3	2	ex 416	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
4	1, 3	mp3an1 1449	. 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
5	4	imp 410	1 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088

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 210  df-an 400  df-3an 1090

This theorem is referenced by:  mp3anr1  1459  facavg  13765  iddvds  15727  isprm7  16161  blometi  28750  mdslmd3i  30279  atcvat2i  30334  chirredlem3  30339  mdsymlem1  30350