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Mirrors > Home > MPE Home > Th. List > Mathboxes > con5VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of con5 43961.
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 43961 is con5VD 44339 without virtual deductions and was automatically
derived from con5VD 44339.
|
Ref | Expression |
---|---|
con5VD | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 44013 | . . . . 5 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) ) | |
2 | biimpr 219 | . . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑)) | |
3 | 1, 2 | e1a 44066 | . . . 4 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) ) |
4 | con3 153 | . . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓)) | |
5 | 3, 4 | e1a 44066 | . . 3 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓) ) |
6 | notnotb 315 | . . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
7 | imbi2 348 | . . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))) | |
8 | 7 | biimprcd 249 | . . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓))) |
9 | 5, 6, 8 | e10 44133 | . 2 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) ) |
10 | 9 | in1 44010 | 1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
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 44009 |
This theorem is referenced by: (None) |
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