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Mirrors > Home > MPE Home > Th. List > Mathboxes > con5VD | Structured version Visualization version GIF version |
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.
|
Ref | Expression |
---|---|
con5VD | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 43325 | . . . . 5 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) ) | |
2 | biimpr 219 | . . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑)) | |
3 | 1, 2 | e1a 43378 | . . . 4 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) ) |
4 | con3 153 | . . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓)) | |
5 | 3, 4 | e1a 43378 | . . 3 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓) ) |
6 | notnotb 314 | . . 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 43445 | . 2 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) ) |
10 | 9 | in1 43322 | 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 43321 |
This theorem is referenced by: (None) |
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