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Theorem el12 39722
 Description: Virtual deduction form of syl2an 590. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
el12.1 (   𝜑   ▶   𝜓   )
el12.2 (   𝜏   ▶   𝜒   )
el12.3 ((𝜓𝜒) → 𝜃)
Assertion
Ref Expression
el12 (   (   𝜑   ,   𝜏   )   ▶   𝜃   )

Proof of Theorem el12
StepHypRef Expression
1 el12.1 . . . 4 (   𝜑   ▶   𝜓   )
21in1 39557 . . 3 (𝜑𝜓)
3 el12.2 . . . 4 (   𝜏   ▶   𝜒   )
43in1 39557 . . 3 (𝜏𝜒)
5 el12.3 . . 3 ((𝜓𝜒) → 𝜃)
62, 4, 5syl2an 590 . 2 ((𝜑𝜏) → 𝜃)
76dfvd2anir 39570 1 (   (   𝜑   ,   𝜏   )   ▶   𝜃   )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385  (   wvd1 39555  (   wvhc2 39566 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 199  df-an 386  df-vd1 39556  df-vhc2 39567 This theorem is referenced by:  elpwgdedVD  39913  sspwimpVD  39915  sspwimpcfVD  39917  suctrALTcfVD  39919
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