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 457	. . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
4	1, 3	mp3anl1 1453	. 2 ⊢ (((𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
5	4	ancoms 457	1 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 394   ∧ 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 395  df-3an 1087

This theorem is referenced by:  vc2OLD  30088  vc0  30094  vcm  30096  nvaddsub4  30177  nvpi  30187  nvge0  30193  ipval3  30229  ipidsq  30230  lnoadd  30278  lnosub  30279  dipsubdir  30368  finorwe  36566