Theorem nfsetrecs 42433

Description: Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)

Hypothesis

Ref	Expression
nfsetrecs.1	|- F/_ x F

Assertion

Ref	Expression
nfsetrecs	|- F/_ xsetrecs ( F )

Proof of Theorem nfsetrecs

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

Step	Hyp	Ref	Expression
1		df-setrecs 42431	. 2 |- setrecs ( F ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
2		nfv 1843	. . . . . . . 8 |- F/ x w C_ y
3		nfv 1843	. . . . . . . . 9 |- F/ x w C_ z
4		nfsetrecs.1	. . . . . . . . . . 11 |- F/_ x F
5		nfcv 2764	. . . . . . . . . . 11 |- F/_ x w
6	4, 5	nffv 6198	. . . . . . . . . 10 |- F/_ x ( F ` w )
7		nfcv 2764	. . . . . . . . . 10 |- F/_ x z
8	6, 7	nfss 3596	. . . . . . . . 9 |- F/ x ( F ` w ) C_ z
9	3, 8	nfim 1825	. . . . . . . 8 |- F/ x ( w C_ z -> ( F ` w ) C_ z )
10	2, 9	nfim 1825	. . . . . . 7 |- F/ x ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) )
11	10	nfal 2153	. . . . . 6 |- F/ x A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) )
12		nfv 1843	. . . . . 6 |- F/ x y C_ z
13	11, 12	nfim 1825	. . . . 5 |- F/ x ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z )
14	13	nfal 2153	. . . 4 |- F/ x A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z )
15	14	nfab 2769	. . 3 |- F/_ x { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
16	15	nfuni 4442	. 2 |- F/_ x U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
17	1, 16	nfcxfr 2762	1 |- F/_ xsetrecs ( F )

Syntax hints:   -> wi 4   A.wal 1481   {cab 2608   F/_wnfc 2751   C_ wss 3574   U.cuni 4436   ` cfv 5888  setrecscsetrecs 42430

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-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-setrecs 42431

This theorem is referenced by: (None)