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Mirrors > Home > MPE Home > Th. List > Mathboxes > con5VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of con5 40863.
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 40863 is con5VD 41241 without virtual deductions and was automatically
derived from con5VD 41241.
|
Ref | Expression |
---|---|
con5VD | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 40915 | . . . . 5 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) ) | |
2 | biimpr 222 | . . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑)) | |
3 | 1, 2 | e1a 40968 | . . . 4 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) ) |
4 | con3 156 | . . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓)) | |
5 | 3, 4 | e1a 40968 | . . 3 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓) ) |
6 | notnotb 317 | . . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
7 | imbi2 351 | . . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))) | |
8 | 7 | biimprcd 252 | . . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓))) |
9 | 5, 6, 8 | e10 41035 | . 2 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) ) |
10 | 9 | in1 40912 | 1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 |
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 209 df-vd1 40911 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |