Theorem nfintd 50998

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 4911	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)}
2		nfv 1948	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1948	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfintd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	4	nfcrd 2920	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴)
6		nfv 1948	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
8	5, 7	nfimd 1928	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
9	3, 8	nfald 2361	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧))
10	2, 9	nfabdw 2947	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∀𝑧(𝑧 ∈ 𝐴 → 𝑦 ∈ 𝑧)})
11	1, 10	nfcxfrd 2925	1 ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1815   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2911  ∩ cint 4910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-int 4911

This theorem is used by: (None)