Theorem syl3an2b 1406

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

Hypotheses

Ref	Expression
syl3an2b.1	⊢ (𝜑 ↔ 𝜒)
syl3an2b.2	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl3an2b	⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Proof of Theorem syl3an2b

Step	Hyp	Ref	Expression
1		syl3an2b.1	. . 3 ⊢ (𝜑 ↔ 𝜒)
2	1	biimpi 219	. 2 ⊢ (𝜑 → 𝜒)
3		syl3an2b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an2 1166	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1089

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 210  df-an 400  df-3an 1091

This theorem is referenced by:  omlimcl  8306  entrfil  8863  cflim2  9877  isdrngd  19792  rintopn  21806  cmpcld  22299  funvtxval0  27106  cusgr0v  27516  2clwwlk2clwwlklem  28429  cgrcomlr  34037  dissneqlem  35248  pmapglb  37521