Theorem syl223anc 1392

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

Hypotheses

Ref	Expression
syl3anc.1	⊢ (𝜑 → 𝜓)
syl3anc.2	⊢ (𝜑 → 𝜒)
syl3anc.3	⊢ (𝜑 → 𝜃)
syl3Xanc.4	⊢ (𝜑 → 𝜏)
syl23anc.5	⊢ (𝜑 → 𝜂)
syl33anc.6	⊢ (𝜑 → 𝜁)
syl133anc.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		syl3anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. . 3 ⊢ (𝜑 → 𝜃)
4		syl3Xanc.4	. . 3 ⊢ (𝜑 → 𝜏)
5	3, 4	jca 514	. 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏))
6		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl133anc.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 1385	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085

This theorem is referenced by:  cdleme17d1  37419  cdlemednpq  37429  cdleme19d  37436  cdleme20aN  37439  cdleme20c  37441  cdleme20f  37444  cdleme20g  37445  cdleme20j  37448  cdleme20l1  37450  cdleme20l2  37451  cdlemky  38056  cdlemkyyN  38092