Theorem syl322anc 1399

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 1393	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ 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 207  df-an 396  df-3an 1088

This theorem is referenced by:  cofcut2d  27893  ax5seglem6  28879  ax5seg  28883  elpaddatriN  39764  paddasslem8  39788  paddasslem12  39792  paddasslem13  39793  pmodlem1  39807  osumcllem5N  39921  pexmidlem2N  39932  cdleme3h  40196  cdleme7ga  40209  cdleme20l  40283  cdleme21ct  40290  cdleme21d  40291  cdleme21e  40292  cdleme26e  40320  cdleme26eALTN  40322  cdleme26fALTN  40323  cdleme26f  40324  cdleme26f2ALTN  40325  cdleme26f2  40326  cdleme39n  40427  cdlemh2  40777  cdlemh  40778  cdlemk12  40811  cdlemk12u  40833  cdlemkfid1N  40882  congsub  42945  mzpcong  42947  jm2.18  42963  jm2.15nn0  42978  jm2.27c  42982