Theorem mp3anl3 1485

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

Hypotheses

Ref	Expression
mp3anl3.1	⊢ 𝜒
mp3anl3.2	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
mp3anl3	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Proof of Theorem mp3anl3

Step	Hyp	Ref	Expression
1		mp3anl3.1	. . 3 ⊢ 𝜒
2		mp3anl3.2	. . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
3	2	ex 417	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏))
4	1, 3	mp3an3 1478	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜏))
5	4	imp 411	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102

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 1104

This theorem is used by:  mp3anr3  1488  hashgt23el  14468  ioombl  25735  nmopadjlem  32452  nmopcoadji  32464  atcvat3i  32759