Theorem syl321anc 1391

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 512	. 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 1383	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 396   ∧ 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 206  df-an 397  df-3an 1088

This theorem is referenced by:  syl322anc  1397  cxple2ad  25880  chordthmlem3  25984  nosupbnd1lem3  33913  nosupbnd1lem4  33914  noinfbnd1lem3  33928  noinfbnd1lem4  33929  4noncolr2  37468  4noncolr1  37469  3atlem5  37501  2lplnj  37634  llnmod2i2  37877  dalawlem11  37895  dalawlem12  37896  cdleme43dN  38506  cdleme4gfv  38521  cdlemeg46nlpq  38531  cdlemg17bq  38687  cdlemg31b0N  38708  cdlemg31b0a  38709  cdlemg31c  38713  cdlemg39  38730  cdlemk47  38963  lincext3  45797