Theorem syl313anc 1302

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)
sylXanc.6	⊢ (𝜑 → 𝜁)
sylXanc.7	⊢ (𝜑 → 𝜎)
syl313anc.8	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)

Assertion

Ref	Expression
syl313anc	⊢ (𝜑 → 𝜌)

Proof of Theorem syl313anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. 2 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. 2 ⊢ (𝜑 → 𝜃)
4		sylXanc.4	. 2 ⊢ (𝜑 → 𝜏)
5		sylXanc.5	. . 3 ⊢ (𝜑 → 𝜂)
6		sylXanc.6	. . 3 ⊢ (𝜑 → 𝜁)
7		sylXanc.7	. . 3 ⊢ (𝜑 → 𝜎)
8	5, 6, 7	3jca 1208	. 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁 ∧ 𝜎))
9		syl313anc.8	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)
10	1, 2, 3, 4, 8, 9	syl311anc 1292	1 ⊢ (𝜑 → 𝜌)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  syl323anc  1308