Theorem syl3an2b 1363

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 206	. 2 ⊢ (𝜑 → 𝜒)
3		syl3an2b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an2 1360	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  omlimcl  7658  cflim2  9085  isdrngd  18772  rintopn  20714  cmpcld  21205  funvtxval0  25897  funvtxval0OLD  25898  cusgr0v  26324  cgrcomlr  32105  dissneqlem  33187  pmapglb  35056