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Theorem nfunid 3642
Description: Deduction version of nfuni 3641. (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 3637 . 2 𝐴 = {𝑦 ∣ ∃𝑧𝐴 𝑦𝑧}
2 nfv 1464 . . 3 𝑦𝜑
3 nfv 1464 . . . 4 𝑧𝜑
4 nfunid.3 . . . 4 (𝜑𝑥𝐴)
5 nfvd 1465 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝑧)
63, 4, 5nfrexdxy 2407 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 𝑦𝑧)
72, 6nfabd 2243 . 2 (𝜑𝑥{𝑦 ∣ ∃𝑧𝐴 𝑦𝑧})
81, 7nfcxfrd 2223 1 (𝜑𝑥 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  {cab 2071  wnfc 2212  wrex 2356   cuni 3635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-uni 3636
This theorem is referenced by:  dfnfc2  3653  nfiotadxy  4945
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