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Theorem nfunid 4918
Description: Deduction version of nfuni 4919. (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 4914 . 2 𝐴 = {𝑦 ∣ ∃𝑧𝐴 𝑦𝑧}
2 nfv 1912 . . 3 𝑦𝜑
3 nfv 1912 . . . 4 𝑧𝜑
4 nfunid.3 . . . 4 (𝜑𝑥𝐴)
5 nfvd 1913 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝑧)
63, 4, 5nfrexdw 3308 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 𝑦𝑧)
72, 6nfabdw 2925 . 2 (𝜑𝑥{𝑦 ∣ ∃𝑧𝐴 𝑦𝑧})
81, 7nfcxfrd 2902 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  {cab 2712  wnfc 2888  wrex 3068   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-uni 4913
This theorem is referenced by:  nfuni  4919  dfnfc2  4934  nfiotadw  6519  nfiotad  6521
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