Theorem nfintd 42420

Description: Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)

Hypothesis

Ref	Expression
nfintd.1	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfintd	|- ( ph -> F/_ x |^| 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 4476	. 2 |- |^| A = { y | A. z ( z e. A -> y e. z ) }
2		nfv 1843	. . 3 |- F/ y ph
3		nfv 1843	. . . 4 |- F/ z ph
4		nfintd.1	. . . . . 6 |- ( ph -> F/_ x A )
5	4	nfcrd 2771	. . . . 5 |- ( ph -> F/ x z e. A )
6		nfv 1843	. . . . . 6 |- F/ x y e. z
7	6	a1i 11	. . . . 5 |- ( ph -> F/ x y e. z )
8	5, 7	nfimd 1823	. . . 4 |- ( ph -> F/ x ( z e. A -> y e. z ) )
9	3, 8	nfald 2165	. . 3 |- ( ph -> F/ x A. z ( z e. A -> y e. z ) )
10	2, 9	nfabd 2785	. 2 |- ( ph -> F/_ x { y | A. z ( z e. A -> y e. z ) } )
11	1, 10	nfcxfrd 2763	1 |- ( ph -> F/_ x |^| A )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708   e. wcel 1990   {cab 2608   F/_wnfc 2751   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-int 4476

This theorem is referenced by: (None)