Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > neeqtri | Structured version Visualization version GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
neeqtr.1 | ⊢ 𝐴 ≠ 𝐵 |
neeqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
neeqtri | ⊢ 𝐴 ≠ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeqtr.1 | . 2 ⊢ 𝐴 ≠ 𝐵 | |
2 | neeqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | neeq2i 3006 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶) |
4 | 1, 3 | mpbi 233 | 1 ⊢ 𝐴 ≠ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2729 df-ne 2941 |
This theorem is referenced by: neeqtrri 3014 sn-0ne2 40097 |
Copyright terms: Public domain | W3C validator |