Theorem nfint 4916

Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Hypothesis

Ref	Expression
nfint.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfint	⊢ Ⅎ𝑥∩ 𝐴

Proof of Theorem nfint

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

Step	Hyp	Ref	Expression
1		dfint2 4908	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfint.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1914	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfralw 3283	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2897	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2889	1 ⊢ Ⅎ𝑥∩ 𝐴

Syntax hints:  {cab 2707  Ⅎwnfc 2876  ∀wral 3044  ∩ cint 4906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-int 4907

This theorem is referenced by:  onminsb  7750  oawordeulem  8495  nnawordex  8578  rankidb  9729  cardmin2  9928  cardaleph  10018  cardmin  10493  sltval2  27544  ldsysgenld  34123  onvf1odlem2  35064  aomclem8  43023  naddwordnexlem4  43363