![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3netr4g | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.) |
Ref | Expression |
---|---|
3netr4g.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
3netr4g.2 | ⊢ 𝐶 = 𝐴 |
3netr4g.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3netr4g | ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3netr4g.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 3netr4g.2 | . . 3 ⊢ 𝐶 = 𝐴 | |
3 | 3netr4g.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
4 | 2, 3 | neeq12i 3007 | . 2 ⊢ (𝐶 ≠ 𝐷 ↔ 𝐴 ≠ 𝐵) |
5 | 1, 4 | sylibr 233 | 1 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-ne 2941 |
This theorem is referenced by: aalioulem2 25709 mapdpglem18 40198 line2x 46926 |
Copyright terms: Public domain | W3C validator |