Theorem syl322anc 1395

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 1389	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084

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 396  df-3an 1086

This theorem is referenced by:  cofcut2d  27758  ax5seglem6  28627  ax5seg  28631  elpaddatriN  39130  paddasslem8  39154  paddasslem12  39158  paddasslem13  39159  pmodlem1  39173  osumcllem5N  39287  pexmidlem2N  39298  cdleme3h  39562  cdleme7ga  39575  cdleme20l  39649  cdleme21ct  39656  cdleme21d  39657  cdleme21e  39658  cdleme26e  39686  cdleme26eALTN  39688  cdleme26fALTN  39689  cdleme26f  39690  cdleme26f2ALTN  39691  cdleme26f2  39692  cdleme39n  39793  cdlemh2  40143  cdlemh  40144  cdlemk12  40177  cdlemk12u  40199  cdlemkfid1N  40248  congsub  42164  mzpcong  42166  jm2.18  42182  jm2.15nn0  42197  jm2.27c  42201