Theorem syl123anc 1386

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 1381	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:  dvfsumlem2  26082  dvfsumlem2OLD  26083  noinfbnd2  27791  atbtwnexOLDN  39430  atbtwnex  39431  osumcllem7N  39945  lhpmcvr5N  40010  cdleme22f2  40330  cdlemefs32sn1aw  40397  cdlemg7aN  40608  cdlemg7N  40609  cdlemg8c  40612  cdlemg8  40614  cdlemg11aq  40621  cdlemg12b  40627  cdlemg12e  40630  cdlemg12g  40632  cdlemg13a  40634  cdlemg15a  40638  cdlemg17e  40648  cdlemg18d  40664  cdlemg19a  40666  cdlemg20  40668  cdlemg22  40670  cdlemg28a  40676  cdlemg29  40688  cdlemg44a  40714  cdlemk34  40893  cdlemn11pre  41193  dihord10  41206  dihord2pre  41208  dihmeetlem17N  41306