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Mirrors > Home > ILE Home > Th. List > necon3bd | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
necon3bd.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) |
Ref | Expression |
---|---|
necon3bd | ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3bd.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) | |
2 | 1 | con3d 593 | . 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵)) |
3 | df-ne 2246 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | syl6ibr 160 | 1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 |
This theorem depends on definitions: df-bi 115 df-ne 2246 |
This theorem is referenced by: nelne1 2335 nelne2 2336 nssne1 3055 nssne2 3056 disjne 3297 difsn 3523 nbrne1 3802 nbrne2 3803 ac6sfi 6379 indpi 6532 zneo 8448 |
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