Theorem syl123anc 1390

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)
syl123anc.7	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)

Assertion

Ref	Expression
syl123anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl123anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. . 3 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. . 3 ⊢ (𝜑 → 𝜃)
4	2, 3	jca 511	. 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃))
5		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl123anc.7	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)
9	1, 4, 5, 6, 7, 8	syl113anc 1385	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  dvfsumlem2  26001  dvfsumlem2OLD  26002  noinfbnd2  27711  atbtwnexOLDN  39820  atbtwnex  39821  osumcllem7N  40335  lhpmcvr5N  40400  cdleme22f2  40720  cdlemefs32sn1aw  40787  cdlemg7aN  40998  cdlemg7N  40999  cdlemg8c  41002  cdlemg8  41004  cdlemg11aq  41011  cdlemg12b  41017  cdlemg12e  41020  cdlemg12g  41022  cdlemg13a  41024  cdlemg15a  41028  cdlemg17e  41038  cdlemg18d  41054  cdlemg19a  41056  cdlemg20  41058  cdlemg22  41060  cdlemg28a  41066  cdlemg29  41078  cdlemg44a  41104  cdlemk34  41283  cdlemn11pre  41583  dihord10  41596  dihord2pre  41598  dihmeetlem17N  41696