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Mirrors > Home > ILE Home > Th. List > necon1abiddc | GIF version |
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.) |
Ref | Expression |
---|---|
necon1abiddc.1 | ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵))) |
Ref | Expression |
---|---|
necon1abiddc | ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon1abiddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵))) | |
2 | 1 | con1biddc 866 | . 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 ↔ 𝜓))) |
3 | df-ne 2335 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 3 | bibi1i 227 | . 2 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ 𝜓)) |
5 | 2, 4 | syl6ibr 161 | 1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 824 = wceq 1342 ≠ wne 2334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-ne 2335 |
This theorem is referenced by: necon2abiddc 2400 |
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