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Mirrors > Home > MPE Home > Th. List > Mathboxes > con5VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of con5 42892.
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 42892 is con5VD 43270 without virtual deductions and was automatically
derived from con5VD 43270.
|
Ref | Expression |
---|---|
con5VD | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 42944 | . . . . 5 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) ) | |
2 | biimpr 219 | . . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑)) | |
3 | 1, 2 | e1a 42997 | . . . 4 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) ) |
4 | con3 153 | . . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓)) | |
5 | 3, 4 | e1a 42997 | . . 3 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓) ) |
6 | notnotb 315 | . . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
7 | imbi2 349 | . . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))) | |
8 | 7 | biimprcd 250 | . . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓))) |
9 | 5, 6, 8 | e10 43064 | . 2 ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) ) |
10 | 9 | in1 42941 | 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 42940 |
This theorem is referenced by: (None) |
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