Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > necon4i | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
Ref | Expression |
---|---|
necon4i.1 | ⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷) |
Ref | Expression |
---|---|
necon4i | ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon4i.1 | . . 3 ⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷) | |
2 | 1 | neneqd 2946 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐶 = 𝐷) |
3 | 2 | necon4ai 2973 | 1 ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ≠ wne 2941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-ne 2942 |
This theorem is referenced by: unixp0 6160 scott0 9526 nn0opthi 13860 |
Copyright terms: Public domain | W3C validator |