Theorem syl3an3b 1428

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 219	. 2 ⊢ (𝜑 → 𝜃)
3		syl3an3b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an3 1181	1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101

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 210  df-an 401  df-3an 1103

This theorem is referenced by:  fnunres2  6638  fresaunres1  6741  fvun2  6963  fvpr2g  7179  nnmsucr  8599  entrfil  9157  enpr2  9976  xrlttr  13156  iccdil  13508  icccntr  13510  hashgt23el  14451  absexpz  15346  nn0rppwr  16609  posglbdg  18459  f1omvdco3  19510  isdrngd  20838  isdrngdOLD  20840  unicld  23164  2ndcdisj2  23575  logrec  26886  cdj3lem3  32699  bnj563  35049  bnj1033  35274  lindsadd  38124  stoweidlem14  46586