Theorem syl3an3b 1401

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

Hypotheses

Ref	Expression
syl3an3b.1	⊢ (𝜑 ↔ 𝜃)
syl3an3b.2	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl3an3b	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Proof of Theorem syl3an3b

Step	Hyp	Ref	Expression
1		syl3an3b.1	. . 3 ⊢ (𝜑 ↔ 𝜃)
2	1	biimpi 218	. 2 ⊢ (𝜑 → 𝜃)
3		syl3an3b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an3 1161	1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by:  fresaunres1  6551  fvun2  6755  nnmsucr  8251  xrlttr  12534  iccdil  12877  icccntr  12879  hashgt23el  13786  absexpz  14665  posglbd  17760  f1omvdco3  18577  isdrngd  19527  unicld  21654  2ndcdisj2  22065  logrec  25341  cdj3lem3  30215  bnj563  32014  bnj1033  32241  lindsadd  34900  nn0rppwr  39202  stoweidlem14  42319