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Theorem syl122anc 1259
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 1255 1 (𝜑𝜁)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 981
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 983
This theorem is referenced by:  divdiv32apd  8902  divcanap5d  8903  divcanap7d  8905  divdivap1d  8908  divdivap2d  8909  seq3coll  11000  cau3lem  11475  summodclem2a  11742  prodmodclem2a  11937  prmind2  12492  divnumden  12568  pceulem  12667  pcqmul  12676  pcqdiv  12680  pcexp  12682  pcaddlem  12712  pcbc  12724  abladdsub4  13700  ablpnpcan  13706  lmodvs1  14128  blss2ps  14928  blss2  14929  blssps  14949  blss  14950  xmeter  14958  lgsdi  15564
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