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Theorem nfel 2202
 Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfnfc.1 𝑥𝐴
nfeq.2 𝑥𝐵
Assertion
Ref Expression
nfel 𝑥 𝐴𝐵

Proof of Theorem nfel
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-clel 2052 . 2 (𝐴𝐵 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝐵))
2 nfcv 2194 . . . . 5 𝑥𝑧
3 nfnfc.1 . . . . 5 𝑥𝐴
42, 3nfeq 2201 . . . 4 𝑥 𝑧 = 𝐴
5 nfeq.2 . . . . 5 𝑥𝐵
65nfcri 2188 . . . 4 𝑥 𝑧𝐵
74, 6nfan 1473 . . 3 𝑥(𝑧 = 𝐴𝑧𝐵)
87nfex 1544 . 2 𝑥𝑧(𝑧 = 𝐴𝑧𝐵)
91, 8nfxfr 1379 1 𝑥 𝐴𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   = wceq 1259  Ⅎwnf 1365  ∃wex 1397   ∈ wcel 1409  Ⅎwnfc 2181 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-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-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183 This theorem is referenced by:  nfel1  2204  nfel2  2206  nfnel  2321  elabgf  2707  elrabf  2718  sbcel12g  2892  nfdisjv  3784  rabxfrd  4228  ffnfvf  5351  elabgft1  10276  elabgf2  10278  bj-rspgt  10284
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