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Theorem exbiriVD 44852
Description: Virtual deduction proof of exbiri 811. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1:: ((𝜑𝜓) → (𝜒𝜃))
2:: (   𝜑   ▶   𝜑   )
3:: (   𝜑   ,   𝜓   ▶   𝜓   )
4:: (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
5:2,1,?: e10 44692 (   𝜑   ▶   (𝜓 → (𝜒𝜃))   )
6:3,5,?: e21 44728 (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )
7:4,6,?: e32 44756 (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
8:7: (   𝜑   ,   𝜓   ▶   (𝜃𝜒)   )
9:8: (   𝜑   ▶   (𝜓 → (𝜃𝜒))   )
qed:9: (𝜑 → (𝜓 → (𝜃𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
exbiriVD.1 ((𝜑𝜓) → (𝜒𝜃))
Assertion
Ref Expression
exbiriVD (𝜑 → (𝜓 → (𝜃𝜒)))

Proof of Theorem exbiriVD
StepHypRef Expression
1 idn3 44613 . . . . 5 (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
2 idn2 44611 . . . . . 6 (   𝜑   ,   𝜓   ▶   𝜓   )
3 idn1 44572 . . . . . . 7 (   𝜑   ▶   𝜑   )
4 exbiriVD.1 . . . . . . 7 ((𝜑𝜓) → (𝜒𝜃))
5 pm3.3 448 . . . . . . . 8 (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜒𝜃))))
65com12 32 . . . . . . 7 (𝜑 → (((𝜑𝜓) → (𝜒𝜃)) → (𝜓 → (𝜒𝜃))))
73, 4, 6e10 44692 . . . . . 6 (   𝜑   ▶   (𝜓 → (𝜒𝜃))   )
8 pm2.27 42 . . . . . 6 (𝜓 → ((𝜓 → (𝜒𝜃)) → (𝜒𝜃)))
92, 7, 8e21 44728 . . . . 5 (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )
10 biimpr 220 . . . . . 6 ((𝜒𝜃) → (𝜃𝜒))
1110com12 32 . . . . 5 (𝜃 → ((𝜒𝜃) → 𝜒))
121, 9, 11e32 44756 . . . 4 (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
1312in3 44607 . . 3 (   𝜑   ,   𝜓   ▶   (𝜃𝜒)   )
1413in2 44603 . 2 (   𝜑   ▶   (𝜓 → (𝜃𝜒))   )
1514in1 44569 1 (𝜑 → (𝜓 → (𝜃𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395
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 207  df-an 396  df-3an 1088  df-vd1 44568  df-vd2 44576  df-vd3 44588
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator