Theorem syl212anc 1376

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)
syl212anc.6	⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂)) → 𝜁)

Assertion

Ref	Expression
syl212anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl212anc

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	. . 3 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6	4, 5	jca 514	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl212anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂)) → 𝜁)
8	1, 2, 3, 6, 7	syl211anc 1372	1 ⊢ (𝜑 → 𝜁)

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:  pntrmax  26142  tglineineq  26431  tglineinteq  26433  paddasslem4  36961  4atexlemu  37202  4atexlemv  37203  cdleme20aN  37447  cdleme20g  37453  cdlemg9a  37770  cdlemg12a  37781  cdlemg17dALTN  37802  cdlemg18b  37817  cdlemg18c  37818