Theorem nfintd 50370

Description: Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)

Hypothesis

Ref	Expression
nfintd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfintd	⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)

Proof of Theorem nfintd

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-int 4917	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}
2		nfv 1941	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1941	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfintd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	4	nfcrd 2925	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴)
6		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
8	5, 7	nfimd 1921	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
9	3, 8	nfald 2367	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
10	2, 9	nfabdw 2952	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)})
11	1, 10	nfcxfrd 2930	1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1565  Ⅎwnf 1810   ∈ wcel 2149  {cab 2747  Ⅎwnfc 2916  ∩ cint 4916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-int 4917

This theorem is referenced by: (None)