Theorem syl3anb 1161

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)

Hypotheses

Ref	Expression
syl3anb.1	⊢ (𝜑 ↔ 𝜓)
syl3anb.2	⊢ (𝜒 ↔ 𝜃)
syl3anb.3	⊢ (𝜏 ↔ 𝜂)
syl3anb.4	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl3anb	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Proof of Theorem syl3anb

Step	Hyp	Ref	Expression
1		syl3anb.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		syl3anb.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
3		syl3anb.3	. . 3 ⊢ (𝜏 ↔ 𝜂)
4	1, 2, 3	3anbi123i 1155	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anb.4	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
6	4, 5	sylbi 217	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

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:  syl3anbr  1162  poxp  8110  infempty  9467  symgsssg  19404  symgfisg  19405  lmodvscl  20791  xrs1mnd  21328  iscnp2  23133  clwwlknccat  29999  slmdvscl  33174  cgr3permute3  36042  cgr3permute1  36043  cgr3permute2  36044  cgr3permute4  36045  cgr3permute5  36046  colinearxfr  36070  grposnOLD  37883  rngunsnply  43165