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Mirrors > Home > MPE Home > Th. List > Mathboxes > aifftbifffaibif | Structured version Visualization version GIF version |
Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.) |
Ref | Expression |
---|---|
aifftbifffaibif.1 | ⊢ (𝜑 ↔ ⊤) |
aifftbifffaibif.2 | ⊢ (𝜓 ↔ ⊥) |
Ref | Expression |
---|---|
aifftbifffaibif | ⊢ ((𝜑 → 𝜓) ↔ ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aifftbifffaibif.1 | . . . . 5 ⊢ (𝜑 ↔ ⊤) | |
2 | 1 | aistia 43931 | . . . 4 ⊢ 𝜑 |
3 | aifftbifffaibif.2 | . . . . 5 ⊢ (𝜓 ↔ ⊥) | |
4 | 3 | aisfina 43932 | . . . 4 ⊢ ¬ 𝜓 |
5 | 2, 4 | pm3.2i 474 | . . 3 ⊢ (𝜑 ∧ ¬ 𝜓) |
6 | annim 407 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)) | |
7 | 6 | biimpi 219 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ ¬ (𝜑 → 𝜓) |
9 | 8 | bifal 1558 | 1 ⊢ ((𝜑 → 𝜓) ↔ ⊥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ⊤wtru 1543 ⊥wfal 1554 |
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 210 df-an 400 df-tru 1545 df-fal 1555 |
This theorem is referenced by: atnaiana 43957 |
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