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Mirrors > Home > ILE Home > Th. List > necon2ai | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.) |
Ref | Expression |
---|---|
necon2ai.1 | ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
Ref | Expression |
---|---|
necon2ai | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2ai.1 | . . 3 ⊢ (𝐴 = 𝐵 → ¬ 𝜑) | |
2 | 1 | con2i 617 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | df-ne 2310 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1332 ≠ wne 2309 |
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 |
This theorem depends on definitions: df-bi 116 df-ne 2310 |
This theorem is referenced by: necon2i 2365 neneqad 2388 intexr 4083 iin0r 4101 tfrlemisucaccv 6230 pm54.43 7063 renepnf 7837 renemnf 7838 lt0ne0d 8299 nnne0 8772 nn0nepnf 9072 hashennn 10558 bj-intexr 13277 |
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