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Theorem nfintd 42930
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 4628 . 2 𝐴 = {𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)}
2 nfv 1992 . . 3 𝑦𝜑
3 nfv 1992 . . . 4 𝑧𝜑
4 nfintd.1 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2909 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
6 nfv 1992 . . . . . 6 𝑥 𝑦𝑧
76a1i 11 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝑧)
85, 7nfimd 1972 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴𝑦𝑧))
93, 8nfald 2310 . . 3 (𝜑 → Ⅎ𝑥𝑧(𝑧𝐴𝑦𝑧))
102, 9nfabd 2923 . 2 (𝜑𝑥{𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)})
111, 10nfcxfrd 2901 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wnf 1857  wcel 2139  {cab 2746  wnfc 2889   cint 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-int 4628
This theorem is referenced by: (None)
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