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Theorem nfne 2312
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfne.1 𝑥𝐴
nfne.2 𝑥𝐵
Assertion
Ref Expression
nfne 𝑥 𝐴𝐵

Proof of Theorem nfne
StepHypRef Expression
1 df-ne 2221 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 nfne.1 . . . 4 𝑥𝐴
3 nfne.2 . . . 4 𝑥𝐵
42, 3nfeq 2201 . . 3 𝑥 𝐴 = 𝐵
54nfn 1564 . 2 𝑥 ¬ 𝐴 = 𝐵
61, 5nfxfr 1379 1 𝑥 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1259  wnf 1365  wnfc 2181  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221
This theorem is referenced by: (None)
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