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Mirrors > Home > ILE Home > Th. List > eqnetrd | GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
eqnetrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqnetrd.2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
eqnetrd | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetrd.2 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
2 | eqnetrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | neeq1d 2365 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
4 | 1, 3 | mpbird 167 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-ne 2348 |
This theorem is referenced by: eqnetrrd 2373 frecabcl 6397 frecsuclem 6404 omp1eomlem 7090 xaddnemnf 9853 xaddnepnf 9854 hashprg 10781 bezoutr1 12026 phibndlem 12208 dfphi2 12212 lgsne0 14310 2sqlem8a 14329 2sqlem8 14330 |
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