Theorem syl311anc 1149

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)

Hypotheses

Ref	Expression
sylXanc.1	⊢ (𝜑 → 𝜓)
sylXanc.2	⊢ (𝜑 → 𝜒)
sylXanc.3	⊢ (𝜑 → 𝜃)
sylXanc.4	⊢ (𝜑 → 𝜏)
sylXanc.5	⊢ (𝜑 → 𝜂)
syl311anc.6	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl311anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl311anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. . 3 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. . 3 ⊢ (𝜑 → 𝜃)
4	1, 2, 3	3jca 1084	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		sylXanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		sylXanc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl311anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜁)
8	4, 5, 6, 7	syl3anc 1135	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  syl312anc  1156  syl321anc  1157  syl313anc  1159  syl331anc  1160