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Theorem syl221anc 1227
 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)
syl221anc.6 (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)
Assertion
Ref Expression
syl221anc (𝜑𝜁)

Proof of Theorem syl221anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . 2 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
4 sylXanc.4 . . 3 (𝜑𝜏)
53, 4jca 304 . 2 (𝜑 → (𝜃𝜏))
6 sylXanc.5 . 2 (𝜑𝜂)
7 syl221anc.6 . 2 (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)
81, 2, 5, 6, 7syl211anc 1222 1 (𝜑𝜁)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116  df-3an 964 This theorem is referenced by:  syl222anc  1232  vtocldf  2737  dmdcanapd  8592  exprecap  10346  xrbdtri  11057  2strbasg  12074  2stropg  12075  cnptoprest  12422  blssps  12610  blss  12611  metequiv2  12679  xmettx  12693
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