Theorem syl322anc 1398

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)
syl133anc.7	⊢ (𝜑 → 𝜎)
syl322anc.8	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎)) → 𝜌)

Assertion

Ref	Expression
syl322anc	⊢ (𝜑 → 𝜌)

Proof of Theorem syl322anc

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	. 2 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7		syl133anc.7	. . 3 ⊢ (𝜑 → 𝜎)
8	6, 7	jca 511	. 2 ⊢ (𝜑 → (𝜁 ∧ 𝜎))
9		syl322anc.8	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎)) → 𝜌)
10	1, 2, 3, 4, 5, 8, 9	syl321anc 1392	1 ⊢ (𝜑 → 𝜌)

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:  cofcut2d  27975  ax5seglem6  28967  ax5seg  28971  elpaddatriN  39760  paddasslem8  39784  paddasslem12  39788  paddasslem13  39789  pmodlem1  39803  osumcllem5N  39917  pexmidlem2N  39928  cdleme3h  40192  cdleme7ga  40205  cdleme20l  40279  cdleme21ct  40286  cdleme21d  40287  cdleme21e  40288  cdleme26e  40316  cdleme26eALTN  40318  cdleme26fALTN  40319  cdleme26f  40320  cdleme26f2ALTN  40321  cdleme26f2  40322  cdleme39n  40423  cdlemh2  40773  cdlemh  40774  cdlemk12  40807  cdlemk12u  40829  cdlemkfid1N  40878  congsub  42927  mzpcong  42929  jm2.18  42945  jm2.15nn0  42960  jm2.27c  42964