Theorem syl3an1b 1430

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 219	. 2 ⊢ (𝜑 → 𝜓)
3		syl3an1b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an1 1181	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103

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 402  df-3an 1105

This theorem is used by:  ovmpoelrn  8083  irrmul  13095  xrlttr  13262  flfneii  24304  padct  33303  crefdf  34473  divrngcl  38871