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Theorem nfeld 2175
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1 (φxA)
nfeqd.2 (φxB)
Assertion
Ref Expression
nfeld (φ → Ⅎx A B)

Proof of Theorem nfeld
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clel 2018 . 2 (A By(y = A y B))
2 nfv 1402 . . 3 yφ
3 nfcvd 2161 . . . . 5 (φxy)
4 nfeqd.1 . . . . 5 (φxA)
53, 4nfeqd 2174 . . . 4 (φ → Ⅎx y = A)
6 nfeqd.2 . . . . 5 (φxB)
76nfcrd 2173 . . . 4 (φ → Ⅎx y B)
85, 7nfand 1442 . . 3 (φ → Ⅎx(y = A y B))
92, 8nfexd 1626 . 2 (φ → Ⅎxy(y = A y B))
101, 9nfxfrd 1344 1 (φ → Ⅎx A B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2015  df-clel 2018  df-nfc 2149
This theorem is referenced by:  nfneld  2283  nfraldxy  2334  nfrexdxy  2335  nfreudxy  2461  nfsbc1d  2757  nfsbcd  2760  nfbrd  3781  nfriotadxy  5400
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