Theorem syl3an1b 1269

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an1b.1	⊢ (𝜑 ↔ 𝜓)
syl3an1b.2	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl3an1b	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Proof of Theorem syl3an1b

Step	Hyp	Ref	Expression
1		syl3an1b.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	biimpi 119	. 2 ⊢ (𝜑 → 𝜓)
3		syl3an1b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an1 1266	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  ovmpoelrn  6186  irrmul  9606  xrlttr  9752