Theorem syl3anb 1157

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 1151	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anb.4	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
6	4, 5	sylbi 219	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

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:  syl3anbr  1158  poxp  7816  infempty  8965  symgsssg  18589  symgfisg  18590  lmodvscl  19645  xrs1mnd  20577  iscnp2  21841  clwwlknccat  27836  slmdvscl  30837  cgr3permute3  33503  cgr3permute1  33504  cgr3permute2  33505  cgr3permute4  33506  cgr3permute5  33507  colinearxfr  33531  grposnOLD  35154  rngunsnply  39766