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Theorem syl5impVD 39413
Description: Virtual deduction proof of syl5imp 39035. 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:: (   (𝜑 → (𝜓𝜒))   ▶   (𝜑 → (𝜓𝜒))   )
2:1,?: e1a 39169 (   (𝜑 → (𝜓𝜒))   ▶   (𝜓 → (𝜑𝜒))   )
3:: (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃𝜓)   )
4:3,2,?: e21 39274 (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃 → (𝜑𝜒))   )
5:4,?: e2 39173 (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜑 → (𝜃𝜒))   )
6:5: (   (𝜑 → (𝜓𝜒))   ▶   ((𝜃 𝜓) → (𝜑 → (𝜃𝜒)))   )
qed:6: ((𝜑 → (𝜓𝜒)) → ((𝜃 𝜓) → (𝜑 → (𝜃𝜒))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
syl5impVD ((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))

Proof of Theorem syl5impVD
StepHypRef Expression
1 idn2 39155 . . . . 5 (   (𝜑 → (𝜓𝜒))   ,   (𝜃𝜓)   ▶   (𝜃𝜓)   )
2 idn1 39107 . . . . . 6 (   (𝜑 → (𝜓𝜒))   ▶   (𝜑 → (𝜓𝜒))   )
3 pm2.04 90 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
42, 3e1a 39169 . . . . 5 (   (𝜑 → (𝜓𝜒))   ▶   (𝜓 → (𝜑𝜒))   )
5 imim1 83 . . . . 5 ((𝜃𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜃 → (𝜑𝜒))))
61, 4, 5e21 39274 . . . 4 (   (𝜑 → (𝜓𝜒))   ,   (𝜃𝜓)   ▶   (𝜃 → (𝜑𝜒))   )
7 pm2.04 90 . . . 4 ((𝜃 → (𝜑𝜒)) → (𝜑 → (𝜃𝜒)))
86, 7e2 39173 . . 3 (   (𝜑 → (𝜓𝜒))   ,   (𝜃𝜓)   ▶   (𝜑 → (𝜃𝜒))   )
98in2 39147 . 2 (   (𝜑 → (𝜓𝜒))   ▶   ((𝜃𝜓) → (𝜑 → (𝜃𝜒)))   )
109in1 39104 1 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4
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 39103  df-vd2 39111
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator