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Theorem nfeld 2315
 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 𝐴𝐵)

Proof of Theorem nfeld
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-clel 2153 . 2 (𝐴𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝐵))
2 nfv 1508 . . 3 𝑦𝜑
3 nfcvd 2300 . . . . 5 (𝜑𝑥𝑦)
4 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeqd 2314 . . . 4 (𝜑 → Ⅎ𝑥 𝑦 = 𝐴)
6 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
76nfcrd 2313 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
85, 7nfand 1548 . . 3 (𝜑 → Ⅎ𝑥(𝑦 = 𝐴𝑦𝐵))
92, 8nfexd 1741 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦 = 𝐴𝑦𝐵))
101, 9nfxfrd 1455 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335  Ⅎwnf 1440  ∃wex 1472   ∈ wcel 2128  Ⅎwnfc 2286 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288 This theorem is referenced by:  nfneld  2430  nfraldw  2489  nfraldxy  2490  nfrexdxy  2491  nfreudxy  2630  nfsbc1d  2953  nfsbcd  2956  sbcrext  3014  nfsbcdw  3065  nfbrd  4009  nfriotadxy  5782  nfixpxy  6655
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