Theorem syl321anc 1395

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 1387	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:  syl322anc  1401  cxple2ad  26702  chordthmlem3  26812  nosupbnd1lem3  27690  nosupbnd1lem4  27691  noinfbnd1lem3  27705  noinfbnd1lem4  27706  4noncolr2  39824  4noncolr1  39825  3atlem5  39857  2lplnj  39990  llnmod2i2  40233  dalawlem11  40251  dalawlem12  40252  cdleme43dN  40862  cdleme4gfv  40877  cdlemeg46nlpq  40887  cdlemg17bq  41043  cdlemg31b0N  41064  cdlemg31b0a  41065  cdlemg31c  41069  cdlemg39  41086  cdlemk47  41319  lincext3  48810