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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 2361 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | 2 | con2i 622 | 1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ≠ wne 2340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-in1 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 df-ne 2341 |
This theorem is referenced by: minel 3476 rzal 3512 difsnb 3723 fin0 6863 0npi 7275 0nsr 7711 renfdisj 7979 nltpnft 9771 ngtmnft 9774 xrrebnd 9776 hashnncl 10730 rennim 10966 pceq0 12275 |
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