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Theorem nfunid 4836
Description: Deduction version of nfuni 4837. (Contributed by NM, 18-Feb-2013.)
Hypothesis
Ref Expression
nfunid.3 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfunid (𝜑𝑥 𝐴)

Proof of Theorem nfunid
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfuni2 4832 . 2 𝐴 = {𝑦 ∣ ∃𝑧𝐴 𝑦𝑧}
2 nfv 1906 . . 3 𝑦𝜑
3 nfv 1906 . . . 4 𝑧𝜑
4 nfunid.3 . . . 4 (𝜑𝑥𝐴)
5 nfvd 1907 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝑧)
63, 4, 5nfrexd 3304 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 𝑦𝑧)
72, 6nfabdw 2997 . 2 (𝜑𝑥{𝑦 ∣ ∃𝑧𝐴 𝑦𝑧})
81, 7nfcxfrd 2973 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  {cab 2796  wnfc 2958  wrex 3136   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-uni 4831
This theorem is referenced by:  nfuni  4837  dfnfc2  4848  nfiotadw  6310  nfiotad  6312
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