Theorem syl3an3b 1407

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 216	. 2 ⊢ (𝜑 → 𝜃)
3		syl3an3b.2	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	2, 3	syl3an3 1165	1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  fnunres2  6631  fresaunres1  6733  fvun2  6953  fvpr2g  7165  nnmsucr  8589  entrfil  9149  enpr2  9955  xrlttr  13100  iccdil  13451  icccntr  13453  hashgt23el  14389  absexpz  15271  nn0rppwr  16531  posglbdg  18374  f1omvdco3  19379  isdrngd  20674  isdrngdOLD  20676  unicld  22933  2ndcdisj2  23344  logrec  26673  cdj3lem3  32367  bnj563  34733  bnj1033  34959  lindsadd  37607  stoweidlem14  46012