Theorem syl123anc 1414

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 521	. 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 1409	1 ⊢ (𝜑 → 𝜎)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  dvfsumlem2  26327  noinfbnd2  28070  atbtwnexOLDN  40472  atbtwnex  40473  osumcllem7N  40987  lhpmcvr5N  41052  cdleme22f2  41372  cdlemefs32sn1aw  41439  cdlemg7aN  41650  cdlemg7N  41651  cdlemg8c  41654  cdlemg8  41656  cdlemg11aq  41663  cdlemg12b  41669  cdlemg12e  41672  cdlemg12g  41674  cdlemg13a  41676  cdlemg15a  41680  cdlemg17e  41690  cdlemg18d  41706  cdlemg19a  41708  cdlemg20  41710  cdlemg22  41712  cdlemg28a  41718  cdlemg29  41730  cdlemg44a  41756  cdlemk34  41935  cdlemn11pre  42235  dihord10  42248  dihord2pre  42250  dihmeetlem17N  42348