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Theorem syl122anc 1247
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
sylXanc.1 (𝜑𝜓)
sylXanc.2 (𝜑𝜒)
sylXanc.3 (𝜑𝜃)
sylXanc.4 (𝜑𝜏)
sylXanc.5 (𝜑𝜂)
syl122anc.6 ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)
Assertion
Ref Expression
syl122anc (𝜑𝜁)

Proof of Theorem syl122anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . 2 (𝜑𝜒)
3 sylXanc.3 . 2 (𝜑𝜃)
4 sylXanc.4 . . 3 (𝜑𝜏)
5 sylXanc.5 . . 3 (𝜑𝜂)
64, 5jca 306 . 2 (𝜑 → (𝜏𝜂))
7 syl122anc.6 . 2 ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)
81, 2, 3, 6, 7syl121anc 1243 1 (𝜑𝜁)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117  df-3an 980
This theorem is referenced by:  divdiv32apd  8773  divcanap5d  8774  divcanap7d  8776  divdivap1d  8779  divdivap2d  8780  seq3coll  10822  cau3lem  11123  summodclem2a  11389  prodmodclem2a  11584  prmind2  12120  divnumden  12196  pceulem  12294  pcqmul  12303  pcqdiv  12307  pcexp  12309  pcaddlem  12338  pcbc  12349  abladdsub4  13117  ablpnpcan  13123  lmodvs1  13406  blss2ps  13909  blss2  13910  blssps  13930  blss  13931  xmeter  13939  lgsdi  14441
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