Theorem syl123anc 1368

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 504	. 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 1363	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069

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 199  df-an 388  df-3an 1071

This theorem is referenced by:  dvfsumlem2  24342  atbtwnexOLDN  36068  atbtwnex  36069  osumcllem7N  36583  lhpmcvr5N  36648  cdleme22f2  36968  cdlemefs32sn1aw  37035  cdlemg7aN  37246  cdlemg7N  37247  cdlemg8c  37250  cdlemg8  37252  cdlemg11aq  37259  cdlemg12b  37265  cdlemg12e  37268  cdlemg12g  37270  cdlemg13a  37272  cdlemg15a  37276  cdlemg17e  37286  cdlemg18d  37302  cdlemg19a  37304  cdlemg20  37306  cdlemg22  37308  cdlemg28a  37314  cdlemg29  37326  cdlemg44a  37352  cdlemk34  37531  cdlemn11pre  37831  dihord10  37844  dihord2pre  37846  dihmeetlem17N  37944