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Theorem con5VD 39633
Description: Virtual deduction proof of con5 39228. 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 39228 is con5VD 39633 without virtual deductions and was automatically derived from con5VD 39633.
 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 39290 . . . . 5 (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2 biimpr 210 . . . . 5 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓𝜑))
31, 2e1a 39352 . . . 4 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
4 con3 149 . . . 4 ((¬ 𝜓𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
53, 4e1a 39352 . . 3 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓)   )
6 notnotb 304 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
7 imbi2 337 . . . 4 ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
87biimprcd 240 . . 3 ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑𝜓)))
95, 6, 8e10 39419 . 2 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
109in1 39287 1 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196 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-vd1 39286 This theorem is referenced by: (None)
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