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Theorem exbirVD 39579
Description: Virtual deduction proof of exbir 39178. 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 39445 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜒𝜃)   )
6:3,5,?: e32 39479 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓), 𝜃   ▶   𝜒   )
7:6: (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜃𝜒)   )
8:7: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜃𝜒))   )
9:8,?: e1a 39346 (   ((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 39334 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜃   )
2 idn1 39284 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
3 idn2 39332 . . . . . . 7 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜑𝜓)   )
4 id 22 . . . . . . 7 (((𝜑𝜓) → (𝜒𝜃)) → ((𝜑𝜓) → (𝜒𝜃)))
52, 3, 4e12 39445 . . . . . 6 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜒𝜃)   )
6 biimpr 210 . . . . . . 7 ((𝜒𝜃) → (𝜃𝜒))
76com12 32 . . . . . 6 (𝜃 → ((𝜒𝜃) → 𝜒))
81, 5, 7e32 39479 . . . . 5 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ,   𝜃   ▶   𝜒   )
98in3 39328 . . . 4 (   ((𝜑𝜓) → (𝜒𝜃))   ,   (𝜑𝜓)   ▶   (𝜃𝜒)   )
109in2 39324 . . 3 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   ((𝜑𝜓) → (𝜃𝜒))   )
11 pm3.3 459 . . 3 (((𝜑𝜓) → (𝜃𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
1210, 11e1a 39346 . 2 (   ((𝜑𝜓) → (𝜒𝜃))   ▶   (𝜑 → (𝜓 → (𝜃𝜒)))   )
1312in1 39281 1 (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383
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-3an 1074  df-vd1 39280  df-vd2 39288  df-vd3 39300
This theorem is referenced by: (None)
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