Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  el2122old Structured version   Visualization version   GIF version

Theorem el2122old 39714
 Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
el2122old.1 (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
el2122old.2 (   𝜓   ▶   𝜃   )
el2122old.3 (   𝜓   ▶   𝜏   )
el2122old.4 ((𝜒𝜃𝜏) → 𝜂)
Assertion
Ref Expression
el2122old (   (   𝜑   ,   𝜓   )   ▶   𝜂   )

Proof of Theorem el2122old
StepHypRef Expression
1 el2122old.1 . . . 4 (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
21dfvd2ani 39569 . . 3 ((𝜑𝜓) → 𝜒)
3 el2122old.2 . . . 4 (   𝜓   ▶   𝜃   )
43in1 39557 . . 3 (𝜓𝜃)
5 el2122old.3 . . . 4 (   𝜓   ▶   𝜏   )
65in1 39557 . . 3 (𝜓𝜏)
7 el2122old.4 . . 3 ((𝜒𝜃𝜏) → 𝜂)
82, 4, 6, 7eel2122old 39713 . 2 ((𝜑𝜓) → 𝜂)
98dfvd2anir 39570 1 (   (   𝜑   ,   𝜓   )   ▶   𝜂   )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1108  (   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-3an 1110  df-vd1 39556  df-vhc2 39567 This theorem is referenced by:  suctrALTcfVD  39919
 Copyright terms: Public domain W3C validator