Theorem syl212anc 1336

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

Hypotheses

Ref	Expression
syl12anc.1	⊢ (𝜑 → 𝜓)
syl12anc.2	⊢ (𝜑 → 𝜒)
syl12anc.3	⊢ (𝜑 → 𝜃)
syl22anc.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		syl12anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl12anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl12anc.3	. 2 ⊢ (𝜑 → 𝜃)
4		syl22anc.4	. . 3 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6	4, 5	jca 554	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl212anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂)) → 𝜁)
8	1, 2, 3, 6, 7	syl211anc 1332	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  pntrmax  25253  tglineineq  25538  tglineinteq  25540  paddasslem4  35109  4atexlemu  35350  4atexlemv  35351  cdleme20aN  35597  cdleme20g  35603  cdlemg9a  35920  cdlemg12a  35931  cdlemg17dALTN  35952  cdlemg18b  35967  cdlemg18c  35968