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Mirrors > Home > ILE Home > Th. List > necon2bi | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.) |
Ref | Expression |
---|---|
necon2bi.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
necon2bi | ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2bi.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 1 | neneqd 2348 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | 2 | con2i 617 | 1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-in1 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 df-ne 2328 |
This theorem is referenced by: minel 3455 rzal 3491 difsnb 3699 fin0 6830 0npi 7233 0nsr 7669 renfdisj 7937 nltpnft 9718 ngtmnft 9721 xrrebnd 9723 hashnncl 10670 rennim 10902 |
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