Theorem syl3an2b 1384

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 208	. 2 ⊢ (𝜑 → 𝜒)
3		syl3an2b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an2 1144	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1068

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 199  df-an 388  df-3an 1070

This theorem is referenced by:  omlimcl  8005  cflim2  9483  isdrngd  19250  rintopn  21221  cmpcld  21714  funvtxval0  26503  cusgr0v  26913  2clwwlk2clwwlklem  27883  cgrcomlr  32986  dissneqlem  34069  pmapglb  36357