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Theorem syl331anc 1397
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)
syl331anc.8 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
Assertion
Ref Expression
syl331anc (𝜑𝜌)

Proof of Theorem syl331anc
StepHypRef Expression
1 syl3anc.1 . 2 (𝜑𝜓)
2 syl3anc.2 . 2 (𝜑𝜒)
3 syl3anc.3 . 2 (𝜑𝜃)
4 syl3Xanc.4 . . 3 (𝜑𝜏)
5 syl23anc.5 . . 3 (𝜑𝜂)
6 syl33anc.6 . . 3 (𝜑𝜁)
74, 5, 63jca 1129 . 2 (𝜑 → (𝜏𝜂𝜁))
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl331anc.8 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
101, 2, 3, 7, 8, 9syl311anc 1386 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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:  syl332anc  1403  syl333anc  1404  qredeu  16695  brbtwn2  28920  3atlem4  39488  3atlem6  39490  llnexchb2  39871  osumcllem9N  39966  cdlemd4  40203  cdleme26fALTN  40364  cdleme26f  40365  cdleme36m  40463  cdlemg17b  40664  cdlemg17h  40670  cdlemk38  40917  cdlemk53b  40958  cdlemkyyN  40964  cdlemk43N  40965
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