Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 589 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | df-ne 2246 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | sylibr 132 | 1 ⊢ (𝜑 → 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: necon2i 2301 neneqad 2324 intexr 3925 iin0r 3943 tfrlemisucaccv 5962 pm54.43 6459 renepnf 7166 renemnf 7167 lt0ne0d 7614 nnne0 8067 bj-intexr 10699 |
Copyright terms: Public domain | W3C validator |