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Mirrors > Home > MPE Home > Th. List > Mathboxes > con5VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of con5 40733.
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 40733 is con5VD 41111 without virtual deductions and was automatically
derived from con5VD 41111.
|
Ref | Expression |
---|---|
con5VD | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 40785 | . . . . 5 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) ) | |
2 | biimpr 221 | . . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑)) | |
3 | 1, 2 | e1a 40838 | . . . 4 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) ) |
4 | con3 156 | . . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓)) | |
5 | 3, 4 | e1a 40838 | . . 3 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓) ) |
6 | notnotb 316 | . . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
7 | imbi2 350 | . . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))) | |
8 | 7 | biimprcd 251 | . . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓))) |
9 | 5, 6, 8 | e10 40905 | . 2 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) ) |
10 | 9 | in1 40782 | 1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 |
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 208 df-vd1 40781 |
This theorem is referenced by: (None) |
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