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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 2739 | . 2 ⊢ 𝐵 = 𝐴 |
3 | eqnetrr.2 | . 2 ⊢ 𝐴 ≠ 𝐶 | |
4 | 2, 3 | eqnetri 3009 | 1 ⊢ 𝐵 ≠ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ≠ wne 2938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-cleq 2722 df-ne 2939 |
This theorem is referenced by: ballotlemii 33800 bj-2upln1upl 36208 sn-0tie0 41614 wallispilem4 45082 |
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