Theorem syl3anr3 1189

Description: A syllogism inference. (Contributed by NM, 23-Aug-2007.)

Hypotheses

Ref	Expression
syl3anr3.1	⊢ (𝜑 → 𝜏)
syl3anr3.2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

Assertion

Ref	Expression
syl3anr3	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜑)) → 𝜂)

Proof of Theorem syl3anr3

Step	Hyp	Ref	Expression
1		syl3anr3.1	. . 3 ⊢ (𝜑 → 𝜏)
2	1	3anim3i 1092	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜑) → (𝜓 ∧ 𝜃 ∧ 𝜏))
3		syl3anr3.2	. 2 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
4	2, 3	sylan2 270	1 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜑)) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by: (None)