Theorem nfint 4917

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 4909	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfint.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfralw 3310	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2929	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2921	1 ⊢ Ⅎ𝑥∩ 𝐴

This proof depends on syntax axioms:  {cab 2739  Ⅎwnfc 2908  ∀wral 3077  ∩ cint 4907

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-int 4908

This theorem is used by:  onminsb  7808  oawordeulem  8562  nnawordex  8646  rankidb  9808  cardmin2  10080  cardaleph  10168  cardmin  10648  ltsval2  28013  ldsysgenld  34793  onvf1odlem2  35883  onprcf1acwevdlem2  35896  vonf1oonfo  35898  aomclem8  44062  naddwordnexlem4  44402