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Theorem con5VD 43651
Description: Virtual deduction proof of con5 43273. 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 43273 is con5VD 43651 without virtual deductions and was automatically derived from con5VD 43651.
1:: (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2:1: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
3:2: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓 )   )
4:: (𝜓 ↔ ¬ ¬ 𝜓)
5:3,4: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
qed:5: ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
(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 43325 . . . . 5 (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2 biimpr 219 . . . . 5 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓𝜑))
31, 2e1a 43378 . . . 4 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
4 con3 153 . . . 4 ((¬ 𝜓𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
53, 4e1a 43378 . . 3 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓)   )
6 notnotb 314 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
7 imbi2 348 . . . 4 ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
87biimprcd 249 . . 3 ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑𝜓)))
95, 6, 8e10 43445 . 2 (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
109in1 43322 1 ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205
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 206  df-vd1 43321
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator