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Theorem syl232anc 1393
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl3anc.1 (𝜑𝜓)
syl3anc.2 (𝜑𝜒)
syl3anc.3 (𝜑𝜃)
syl3Xanc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl133anc.7 (𝜑𝜎)
syl232anc.8 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)
Assertion
Ref Expression
syl232anc (𝜑𝜌)

Proof of Theorem syl232anc
StepHypRef Expression
1 syl3anc.1 . 2 (𝜑𝜓)
2 syl3anc.2 . 2 (𝜑𝜒)
3 syl3anc.3 . 2 (𝜑𝜃)
4 syl3Xanc.4 . 2 (𝜑𝜏)
5 syl23anc.5 . 2 (𝜑𝜂)
6 syl33anc.6 . . 3 (𝜑𝜁)
7 syl133anc.7 . . 3 (𝜑𝜎)
86, 7jca 514 . 2 (𝜑 → (𝜁𝜎))
9 syl232anc.8 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)
101, 2, 3, 4, 5, 8, 9syl231anc 1386 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083
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 209  df-an 399  df-3an 1085
This theorem is referenced by:  ax5seg  26718  cdleme20d  37442  cdleme22cN  37472  cdleme27a  37497
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