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Theorem nfunid 4409
Description: Deduction version of nfuni 4408. (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 4404 . 2 𝐴 = {𝑦 ∣ ∃𝑧𝐴 𝑦𝑧}
2 nfv 1840 . . 3 𝑦𝜑
3 nfv 1840 . . . 4 𝑧𝜑
4 nfunid.3 . . . 4 (𝜑𝑥𝐴)
5 nfvd 1841 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝑧)
63, 4, 5nfrexd 3000 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 𝑦𝑧)
72, 6nfabd 2781 . 2 (𝜑𝑥{𝑦 ∣ ∃𝑧𝐴 𝑦𝑧})
81, 7nfcxfrd 2760 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  {cab 2607  wnfc 2748  wrex 2908   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-uni 4403
This theorem is referenced by:  dfnfc2  4420  dfnfc2OLD  4421  nfiotad  5813
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