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Theorem nfintd 42218
 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 4411 . 2 𝐴 = {𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)}
2 nfv 1830 . . 3 𝑦𝜑
3 nfv 1830 . . . 4 𝑧𝜑
4 nfcv 2751 . . . . . . 7 𝑥𝑧
54a1i 11 . . . . . 6 (𝜑𝑥𝑧)
6 nfintd.1 . . . . . 6 (𝜑𝑥𝐴)
75, 6nfeld 2759 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
8 nfv 1830 . . . . . 6 𝑥 𝑦𝑧
98a1i 11 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝑧)
107, 9nfimd 1812 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴𝑦𝑧))
113, 10nfald 2151 . . 3 (𝜑 → Ⅎ𝑥𝑧(𝑧𝐴𝑦𝑧))
122, 11nfabd 2771 . 2 (𝜑𝑥{𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)})
131, 12nfcxfrd 2750 1 (𝜑𝑥 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-int 4411 This theorem is referenced by: (None)
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