Theorem syl321anc 1394

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)
syl321anc.7	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ 𝜁) → 𝜎)

Assertion

Ref	Expression
syl321anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl321anc

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 511	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl321anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ 𝜁) → 𝜎)
9	1, 2, 3, 6, 7, 8	syl311anc 1386	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  syl322anc  1400  cxple2ad  26690  chordthmlem3  26800  nosupbnd1lem3  27678  nosupbnd1lem4  27679  noinfbnd1lem3  27693  noinfbnd1lem4  27694  4noncolr2  39710  4noncolr1  39711  3atlem5  39743  2lplnj  39876  llnmod2i2  40119  dalawlem11  40137  dalawlem12  40138  cdleme43dN  40748  cdleme4gfv  40763  cdlemeg46nlpq  40773  cdlemg17bq  40929  cdlemg31b0N  40950  cdlemg31b0a  40951  cdlemg31c  40955  cdlemg39  40972  cdlemk47  41205  lincext3  48698