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Theorem syl223anc 1502
 Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl12anc.1 (𝜑𝜓)
syl12anc.2 (𝜑𝜒)
syl12anc.3 (𝜑𝜃)
syl22anc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl133anc.7 (𝜑𝜎)
syl223anc.8 (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)
Assertion
Ref Expression
syl223anc (𝜑𝜌)

Proof of Theorem syl223anc
StepHypRef Expression
1 syl12anc.1 . 2 (𝜑𝜓)
2 syl12anc.2 . 2 (𝜑𝜒)
3 syl12anc.3 . . 3 (𝜑𝜃)
4 syl22anc.4 . . 3 (𝜑𝜏)
53, 4jca 501 . 2 (𝜑 → (𝜃𝜏))
6 syl23anc.5 . 2 (𝜑𝜂)
7 syl33anc.6 . 2 (𝜑𝜁)
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl223anc.8 . 2 (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)
101, 2, 5, 6, 7, 8, 9syl213anc 1495 1 (𝜑𝜌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071 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 197  df-an 383  df-3an 1073 This theorem is referenced by:  cdleme17d1  36098  cdlemednpq  36108  cdleme19d  36115  cdleme20aN  36118  cdleme20c  36120  cdleme20f  36123  cdleme20g  36124  cdleme20j  36127  cdleme20l1  36129  cdleme20l2  36130  cdlemky  36735  cdlemkyyN  36771
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