Theorem mp3anl1 1484

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 418	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
4	1, 3	mp3an1 1477	. 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
5	4	imp 412	1 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ 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 402  df-3an 1105

This theorem is used by:  mp3anr1  1487  domssl  9003  domssr  9004  rexdif1en  9154  dif1en  9155  facavg  14412  iddvds  16406  isprm7  16846  blometi  31338  mdslmd3i  32867  atcvat2i  32922  chirredlem3  32927  mdsymlem1  32938