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Theorem el12 39447
Description: Virtual deduction form of syl2an 495. (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 39281 . . 3 (𝜑𝜓)
3 el12.2 . . . 4 (   𝜏   ▶   𝜒   )
43in1 39281 . . 3 (𝜏𝜒)
5 el12.3 . . 3 ((𝜓𝜒) → 𝜃)
62, 4, 5syl2an 495 . 2 ((𝜑𝜏) → 𝜃)
76dfvd2anir 39294 1 (   (   𝜑   ,   𝜏   )   ▶   𝜃   )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  (   wvd1 39279  (   wvhc2 39290
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 385  df-vd1 39280  df-vhc2 39291
This theorem is referenced by:  elpwgdedVD  39644  sspwimpVD  39646  sspwimpcfVD  39648  suctrALTcfVD  39650
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