Theorem mp3anr1 1486

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 463	. . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
4	1, 3	mp3anl1 1483	. 2 ⊢ (((𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏)
5	4	ancoms 463	1 ⊢ ((𝜑 ∧ (𝜒 ∧ 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:  vc2OLD  30931  vc0  30937  vcm  30939  nvaddsub4  31020  nvpi  31030  nvge0  31036  ipval3  31072  ipidsq  31073  lnoadd  31121  lnosub  31122  dipsubdir  31211  finorwe  38056