Theorem mp3anr1 1456

Description: An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)

Hypotheses

Ref	Expression
mp3anr1.1	⊢ 𝜓
mp3anr1.2	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)

Assertion

Ref	Expression
mp3anr1	⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Proof of Theorem mp3anr1

Step	Hyp	Ref	Expression
1		mp3anr1.1	. . 3 ⊢ 𝜓
2		mp3anr1.2	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)
3	2	ancoms 458	. . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
4	1, 3	mp3anl1 1453	. 2 ⊢ (((𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
5	4	ancoms 458	1 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 396  df-3an 1087

This theorem is referenced by:  vc2OLD  28831  vc0  28837  vcm  28839  nvaddsub4  28920  nvpi  28930  nvge0  28936  ipval3  28972  ipidsq  28973  lnoadd  29021  lnosub  29022  dipsubdir  29111  finorwe  35480