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Theorem con5VD 40659
Description: Virtual deduction proof of con5 40281. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con5 40281 is con5VD 40659 without virtual deductions and was automatically derived from con5VD 40659.
 1:: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   ) 2:1: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   ) 3:2: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓 )   ) 4:: ⊢ (𝜓 ↔ ¬ ¬ 𝜓) 5:3,4: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   ) qed:5: ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con5VD ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))

Proof of Theorem con5VD
StepHypRef Expression
1 idn1 40333 . . . . 5 (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2 biimpr 212 . . . . 5 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓𝜑))
31, 2e1a 40386 . . . 4 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
4 con3 151 . . . 4 ((¬ 𝜓𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
53, 4e1a 40386 . . 3 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓)   )
6 notnotb 307 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
7 imbi2 341 . . . 4 ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
87biimprcd 242 . . 3 ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑𝜓)))
95, 6, 8e10 40453 . 2 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
109in1 40330 1 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198 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-vd1 40329 This theorem is referenced by: (None)
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