Theorem mp3anr2 1458

Description: An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)

Hypotheses

Ref	Expression
mp3anr2.1	⊢ 𝜒
mp3anr2.2	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)

Assertion

Ref	Expression
mp3anr2	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏)

Proof of Theorem mp3anr2

Step	Hyp	Ref	Expression
1		mp3anr2.1	. . 3 ⊢ 𝜒
2		mp3anr2.2	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)
3	2	ancoms 459	. . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
4	1, 3	mp3anl2 1455	. 2 ⊢ (((𝜓 ∧ 𝜃) ∧ 𝜑) → 𝜏)
5	4	ancoms 459	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 396   ∧ 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 397  df-3an 1088

This theorem is referenced by:  mulgp1  18736  vcz  28937  nvmdi  29010