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Mirrors > Home > ILE Home > Th. List > necon2bd | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.) |
Ref | Expression |
---|---|
necon2bd.1 | ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
Ref | Expression |
---|---|
necon2bd | ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2bd.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) | |
2 | df-ne 2348 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | syl6ib 161 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵)) |
4 | 3 | con2d 624 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-in1 614 ax-in2 615 |
This theorem depends on definitions: df-bi 117 df-ne 2348 |
This theorem is referenced by: disjiun 3995 map0g 6682 nneo 9342 zeo2 9345 bezoutr1 12014 coprm 12124 sqrt2irr 12142 dfphi2 12200 bj-charfunr 14215 nconstwlpolem 14465 |
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