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Theorem nfeld 2802
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 2647 . 2 (𝐴𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝐵))
2 nfv 1883 . . 3 𝑦𝜑
3 nfcvd 2794 . . . . 5 (𝜑𝑥𝑦)
4 nfeqd.1 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeqd 2801 . . . 4 (𝜑 → Ⅎ𝑥 𝑦 = 𝐴)
6 nfeqd.2 . . . . 5 (𝜑𝑥𝐵)
76nfcrd 2800 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐵)
85, 7nfand 1866 . . 3 (𝜑 → Ⅎ𝑥(𝑦 = 𝐴𝑦𝐵))
92, 8nfexd 2203 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦 = 𝐴𝑦𝐵))
101, 9nfxfrd 1820 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by:  nfel  2806  nfneld  2934  nfrald  2973  ralcom2  3133  nfreud  3141  nfrmod  3142  nfrmo  3144  nfsbc1d  3486  nfsbcd  3489  sbcrext  3544  sbcrextOLD  3545  nfdisj  4664  nfbrd  4731  nfriotad  6659  nfixp  7969  axrepndlem2  9453  axrepnd  9454  axunnd  9456  axpowndlem2  9458  axpowndlem3  9459  axpowndlem4  9460  axpownd  9461  axregndlem2  9463  axinfndlem1  9465  axinfnd  9466  axacndlem4  9470  axacndlem5  9471  axacnd  9472
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