Theorem syl3an3b 1530

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 208	. 2 ⊢ (𝜑 → 𝜃)
3		syl3an3b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an3 1211	1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1113

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 387  df-3an 1115

This theorem is referenced by:  fresaunres1  6314  fvun2  6517  nnmsucr  7972  xrlttr  12259  iccdil  12603  icccntr  12605  absexpz  14422  posglbd  17503  f1omvdco3  18219  isdrngd  19128  unicld  21221  2ndcdisj2  21631  logrec  24903  cdj3lem3  29852  bnj563  31359  bnj1033  31583  lindsadd  33946  stoweidlem14  41025