Theorem nfint 3776

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 3768	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfint.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1508	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfralxy 2469	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2284	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2276	1 ⊢ Ⅎ𝑥∩ 𝐴

Syntax hints:  {cab 2123  Ⅎwnfc 2266  ∀wral 2414  ∩ cint 3766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-int 3767

This theorem is referenced by: (None)