Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqnetrri | Structured version Visualization version GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
eqnetrr.1 | ⊢ 𝐴 = 𝐵 |
eqnetrr.2 | ⊢ 𝐴 ≠ 𝐶 |
Ref | Expression |
---|---|
eqnetrri | ⊢ 𝐵 ≠ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetrr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eqcomi 2830 | . 2 ⊢ 𝐵 = 𝐴 |
3 | eqnetrr.2 | . 2 ⊢ 𝐴 ≠ 𝐶 | |
4 | 2, 3 | eqnetri 3086 | 1 ⊢ 𝐵 ≠ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ≠ wne 3016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2814 df-ne 3017 |
This theorem is referenced by: ballotlemii 31661 bj-2upln1upl 34234 wallispilem4 42234 |
Copyright terms: Public domain | W3C validator |