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Theorem exbirVD 39838
Description: Virtual deduction proof of exbir 39453. 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.
1:: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜒𝜃))   )
2:: (   ((𝜑𝜓) → (𝜒𝜃))   ,    (𝜑𝜓)   ▶   (𝜑𝜓)   )
3:: (   ((𝜑𝜓) → (𝜒𝜃))   ,    (𝜑𝜓), 𝜃   ▶   𝜃   )
5:1,2,?: e12 39709 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜒𝜃)   )
6:3,5,?: e32 39743 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓), 𝜃   ▶   𝜒   )
7:6: (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜃𝜒)   )
8:7: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜃𝜒))   )
9:8,?: e1a 39611 (   ((𝜑𝜓) → (𝜒 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃𝜒)))   )
qed:9: (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
exbirVD (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Proof of Theorem exbirVD
StepHypRef Expression
1 idn3 39599 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜃   )
2 idn1 39549 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
3 idn2 39597 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜑𝜓)   )
4 id 22 . . . . . . 7 (((𝜑𝜓) → (𝜒𝜃)) → ((𝜑𝜓) → (𝜒𝜃)))
52, 3, 4e12 39709 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜒𝜃)   )
6 biimpr 212 . . . . . . 7 ((𝜒𝜃) → (𝜃𝜒))
76com12 32 . . . . . 6 (𝜃 → ((𝜒𝜃) → 𝜒))
81, 5, 7e32 39743 . . . . 5 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜒   )
98in3 39593 . . . 4 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜃𝜒)   )
109in2 39589 . . 3 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜃𝜒))   )
11 pm3.3 440 . . 3 (((𝜑𝜓) → (𝜃𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
1210, 11e1a 39611 . 2 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   (𝜑 → (𝜓 → (𝜃𝜒)))   )
1312in1 39546 1 (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385
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 39545  df-vd2 39553  df-vd3 39565
This theorem is referenced by: (None)
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