Theorem syl3anb 1179

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 1173	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anb.4	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
6	4, 5	sylbi 220	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  syl3anbr  1180  poxp  8138  infempty  9494  symgsssg  19674  symgfisg  19675  lmodvscl  21146  xrs1mnd  21739  iscnp2  23550  elreno2  28874  clwwlknccat  30647  slmdvscl  33768  cgr3permute3  36792  cgr3permute1  36793  cgr3permute2  36794  cgr3permute4  36795  cgr3permute5  36796  colinearxfr  36820  grposnOLD  38796  rngunsnply  44155