Theorem syl312anc 1372

Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-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)
syl312anc.7	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎)

Assertion

Ref	Expression
syl312anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl312anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. 2 ⊢ (𝜑 → 𝜃)
4		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7	5, 6	jca 504	. 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁))
8		syl312anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎)
9	1, 2, 3, 4, 7, 8	syl311anc 1365	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069

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 199  df-an 388  df-3an 1071

This theorem is referenced by:  pythagtriplem19  16024  cdleme27cl  36984  cdlemefs27cl  37031  cdleme32fvcl  37058  cdlemg16ALTN  37276  cdlemg27a  37310  cdlemg31c  37317  cdlemg39  37334  cdlemk11ta  37547  cdlemk19ylem  37548  cdlemk11tc  37563  cdlemk45  37565  dihmeetlem12N  37936  dihjatc  38035