Theorem syl3an2b 1430

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 1181	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1102

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 401  df-3an 1104

This theorem is used by:  omlimcl  8561  entrfil  9167  cflim2  10253  isdrngd  20879  isdrngdOLD  20881  rintopn  23077  cmpcld  23570  funvtxval0  29376  cusgr0v  29789  2clwwlk2clwwlklem  30708  cgrcomlr  36498  dissneqlem  38014  pmapglb  40572