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Theorem con5VD 38158
Description: Virtual deduction proof of con5 37749. 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 37749 is con5VD 38158 without virtual deductions and was automatically derived from con5VD 38158.
 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 37811 . . . . 5 (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2 biimpr 209 . . . . 5 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓𝜑))
31, 2e1a 37873 . . . 4 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
4 con3 148 . . . 4 ((¬ 𝜓𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
53, 4e1a 37873 . . 3 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓)   )
6 notnotb 303 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
7 imbi2 337 . . . 4 ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
87biimprcd 239 . . 3 ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑𝜓)))
95, 6, 8e10 37940 . 2 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
109in1 37808 1 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195 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 196  df-vd1 37807 This theorem is referenced by: (None)
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