![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1541 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-ne 2941 |
This theorem is referenced by: xpcoidgend 14924 fclsfnflim 23538 ptcmplem2 23564 vieta1lem1 25830 vieta1lem2 25831 fsuppcurry1 31988 fsuppcurry2 31989 signsvfpn 33665 signsvfnn 33666 finxpreclem2 36357 finxp1o 36359 cdleme3h 39192 cdleme7ga 39205 imo72b2lem1 43003 fourierdlem42 44944 |
Copyright terms: Public domain | W3C validator |