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Theorem nfunid 4579
Description: Deduction version of nfuni 4578. (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 4574 . 2 𝐴 = {𝑦 ∣ ∃𝑧𝐴 𝑦𝑧}
2 nfv 1994 . . 3 𝑦𝜑
3 nfv 1994 . . . 4 𝑧𝜑
4 nfunid.3 . . . 4 (𝜑𝑥𝐴)
5 nfvd 1995 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝑧)
63, 4, 5nfrexd 3153 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 𝑦𝑧)
72, 6nfabd 2933 . 2 (𝜑𝑥{𝑦 ∣ ∃𝑧𝐴 𝑦𝑧})
81, 7nfcxfrd 2911 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  {cab 2756  wnfc 2899  wrex 3061   cuni 4572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-uni 4573
This theorem is referenced by:  dfnfc2  4590  nfiotad  5997
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