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Mirrors > Home > MPE Home > Th. List > eqnetrrid | Structured version Visualization version GIF version |
Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
Ref | Expression |
---|---|
eqnetrrid.1 | ⊢ 𝐵 = 𝐴 |
eqnetrrid.2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
eqnetrrid | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetrrid.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqnetrrid.2 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
4 | 2, 3 | eqnetrrd 3009 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
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: xpcoidgend 14866 fclsfnflim 23394 ptcmplem2 23420 vieta1lem1 25686 vieta1lem2 25687 fsuppcurry1 31689 fsuppcurry2 31690 signsvfpn 33254 signsvfnn 33255 finxpreclem2 35907 finxp1o 35909 cdleme3h 38744 cdleme7ga 38757 imo72b2lem1 42530 fourierdlem42 44476 |
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