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Theorem syl213anc 1342
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)
syl213anc.7 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)
Assertion
Ref Expression
syl213anc (𝜑𝜎)

Proof of Theorem syl213anc
StepHypRef Expression
1 syl12anc.1 . . 3 (𝜑𝜓)
2 syl12anc.2 . . 3 (𝜑𝜒)
31, 2jca 554 . 2 (𝜑 → (𝜓𝜒))
4 syl12anc.3 . 2 (𝜑𝜃)
5 syl22anc.4 . 2 (𝜑𝜏)
6 syl23anc.5 . 2 (𝜑𝜂)
7 syl33anc.6 . 2 (𝜑𝜁)
8 syl213anc.7 . 2 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)
93, 4, 5, 6, 7, 8syl113anc 1335 1 (𝜑𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036
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 386  df-3an 1038
This theorem is referenced by:  syl223anc  1349  decpmatmul  20491
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