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Theorem nfintd 47966
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 4942 . 2 𝐴 = {𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)}
2 nfv 1909 . . 3 𝑦𝜑
3 nfv 1909 . . . 4 𝑧𝜑
4 nfintd.1 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2884 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
6 nfv 1909 . . . . . 6 𝑥 𝑦𝑧
76a1i 11 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝑧)
85, 7nfimd 1889 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴𝑦𝑧))
93, 8nfald 2313 . . 3 (𝜑 → Ⅎ𝑥𝑧(𝑧𝐴𝑦𝑧))
102, 9nfabdw 2918 . 2 (𝜑𝑥{𝑦 ∣ ∀𝑧(𝑧𝐴𝑦𝑧)})
111, 10nfcxfrd 2894 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wnf 1777  wcel 2098  {cab 2701  wnfc 2875   cint 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-int 4942
This theorem is referenced by: (None)
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