Theorem nfint 4924

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 4916	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfint.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfralw 3314	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2933	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2925	1 ⊢ Ⅎ𝑥∩ 𝐴

This proof depends on syntax axioms:  {cab 2743  Ⅎwnfc 2912  ∀wral 3081  ∩ cint 4914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-int 4915

This theorem is used by:  onminsb  7799  oawordeulem  8545  nnawordex  8629  rankidb  9779  cardmin2  10001  cardaleph  10089  cardmin  10565  ltsval2  27873  ldsysgenld  34617  onvf1odlem2  35647  vonf1oonfo  35658  aomclem8  43848  naddwordnexlem4  44188