Theorem syl3an1b 1405

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 216	. 2 ⊢ (𝜑 → 𝜓)
3		syl3an1b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an1 1164	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  ovmpoelrn  8097  dif1enOLD  9202  irrmul  13016  xrlttr  13182  flfneii  24000  padct  32731  crefdf  33847  divrngcl  37964