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Theorem nfintd 43525
 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 4711 . 2 𝐴 = {𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)}
2 nfv 1957 . . 3 𝑦𝜑
3 nfv 1957 . . . 4 𝑧𝜑
4 nfintd.1 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2927 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
6 nfv 1957 . . . . . 6 𝑥 𝑦𝑧
76a1i 11 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝑧)
85, 7nfimd 1940 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴𝑦𝑧))
93, 8nfald 2304 . . 3 (𝜑 → Ⅎ𝑥𝑧(𝑧𝐴𝑦𝑧))
102, 9nfabd 2954 . 2 (𝜑𝑥{𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)})
111, 10nfcxfrd 2933 1 (𝜑𝑥 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1599  Ⅎwnf 1827   ∈ wcel 2107  {cab 2763  Ⅎwnfc 2919  ∩ cint 4710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-int 4711 This theorem is referenced by: (None)
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