Theorem syl332anc 1264

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)
sylXanc.8	⊢ (𝜑 → 𝜌)
syl332anc.9	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇)

Assertion

Ref	Expression
syl332anc	⊢ (𝜑 → 𝜇)

Proof of Theorem syl332anc

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	. 2 ⊢ (𝜑 → 𝜂)
6		sylXanc.6	. 2 ⊢ (𝜑 → 𝜁)
7		sylXanc.7	. . 3 ⊢ (𝜑 → 𝜎)
8		sylXanc.8	. . 3 ⊢ (𝜑 → 𝜌)
9	7, 8	jca 304	. 2 ⊢ (𝜑 → (𝜎 ∧ 𝜌))
10		syl332anc.9	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇)
11	1, 2, 3, 4, 5, 6, 9, 10	syl331anc 1258	1 ⊢ (𝜑 → 𝜇)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by: (None)