Theorem nfintd 48904

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 4952	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}
2		nfv 1912	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1912	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfintd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	4	nfcrd 2897	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴)
6		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
8	5, 7	nfimd 1892	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
9	3, 8	nfald 2327	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
10	2, 9	nfabdw 2925	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)})
11	1, 10	nfcxfrd 2902	1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1780   ∈ wcel 2106  {cab 2712  Ⅎwnfc 2888  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-int 4952

This theorem is referenced by: (None)