Theorem syl223anc 1254

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)
syl223anc.8	⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)

Assertion

Ref	Expression
syl223anc	⊢ (𝜑 → 𝜌)

Proof of Theorem syl223anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. 2 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. . 3 ⊢ (𝜑 → 𝜃)
4		sylXanc.4	. . 3 ⊢ (𝜑 → 𝜏)
5	3, 4	jca 304	. 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏))
6		sylXanc.5	. 2 ⊢ (𝜑 → 𝜂)
7		sylXanc.6	. 2 ⊢ (𝜑 → 𝜁)
8		sylXanc.7	. 2 ⊢ (𝜑 → 𝜎)
9		syl223anc.8	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)
10	1, 2, 5, 6, 7, 8, 9	syl213anc 1247	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968

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 970

This theorem is referenced by: (None)