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Theorem nfintd 42185
Description: Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
Hypothesis
Ref Expression
nfintd.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfintd (𝜑𝑥 𝐴)

Proof of Theorem nfintd
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-int 4467 . 2 𝐴 = {𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)}
2 nfv 1841 . . 3 𝑦𝜑
3 nfv 1841 . . . 4 𝑧𝜑
4 nfintd.1 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2768 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
6 nfv 1841 . . . . . 6 𝑥 𝑦𝑧
76a1i 11 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝑧)
85, 7nfimd 1821 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴𝑦𝑧))
93, 8nfald 2163 . . 3 (𝜑 → Ⅎ𝑥𝑧(𝑧𝐴𝑦𝑧))
102, 9nfabd 2782 . 2 (𝜑𝑥{𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)})
111, 10nfcxfrd 2761 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wnf 1706  wcel 1988  {cab 2606  wnfc 2749   cint 4466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-int 4467
This theorem is referenced by: (None)
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