Theorem syl3anr2 1379

Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.)

Hypotheses

Ref	Expression
syl3anr2.1	⊢ (𝜑 → 𝜃)
syl3anr2.2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

Assertion

Ref	Expression
syl3anr2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)

Proof of Theorem syl3anr2

Step	Hyp	Ref	Expression
1		syl3anr2.1	. . 3 ⊢ (𝜑 → 𝜃)
2		syl3anr2.2	. . . 4 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
3	2	ancoms 469	. . 3 ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜏) ∧ 𝜒) → 𝜂)
4	1, 3	syl3anl2 1375	. 2 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜂)
5	4	ancoms 469	1 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  mulgsubdir  17582  dipassr2  27702