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Mirrors > Home > ILE Home > Th. List > neeq2 | GIF version |
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) |
Ref | Expression |
---|---|
neeq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2180 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | |
2 | 1 | notbid 662 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵)) |
3 | df-ne 2341 | . 2 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴) | |
4 | df-ne 2341 | . 2 ⊢ (𝐶 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐵) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1348 ≠ wne 2340 |
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 609 ax-in2 610 ax-5 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 df-ne 2341 |
This theorem is referenced by: neeq2i 2356 neeq2d 2359 disji2 3982 fodjuomnilemdc 7120 xrlttri3 9754 neapmkv 14099 |
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