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Theorem nfeld 2335
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 2173 . 2 (𝐴𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝐵))
2 nfv 1528 . . 3 𝑦𝜑
3 nfcvd 2320 . . . . 5 (𝜑𝑥𝑦)
4 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeqd 2334 . . . 4 (𝜑 → Ⅎ𝑥 𝑦 = 𝐴)
6 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
76nfcrd 2333 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
85, 7nfand 1568 . . 3 (𝜑 → Ⅎ𝑥(𝑦 = 𝐴𝑦𝐵))
92, 8nfexd 1761 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦 = 𝐴𝑦𝐵))
101, 9nfxfrd 1475 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wnf 1460  wex 1492  wcel 2148  wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308
This theorem is referenced by:  nfneld  2450  nfraldw  2509  nfraldxy  2510  nfrexdxy  2511  nfreudxy  2650  nfsbc1d  2979  nfsbcd  2982  sbcrext  3040  nfsbcdw  3091  nfbrd  4047  nfriotadxy  5836  nfixpxy  6714
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