Theorem nfraldya 2541

Description: Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2539 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldya.2	|- F/ y ph
nfraldya.3	|- ( ph -> F/_ x A )
nfraldya.4	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfraldya	|- ( ph -> F/ x A. y e. A ps )

Distinct variable group:   y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x)

Proof of Theorem nfraldya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2489	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
2		sbim 1981	. . . . . 6 |- ( [ z / y ] ( y e. A -> ps ) <-> ( [ z / y ] y e. A -> [ z / y ] ps ) )
3		clelsb1 2310	. . . . . . 7 |- ( [ z / y ] y e. A <-> z e. A )
4	3	imbi1i 238	. . . . . 6 |- ( ( [ z / y ] y e. A -> [ z / y ] ps ) <-> ( z e. A -> [ z / y ] ps ) )
5	2, 4	bitri 184	. . . . 5 |- ( [ z / y ] ( y e. A -> ps ) <-> ( z e. A -> [ z / y ] ps ) )
6	5	albii 1493	. . . 4 |- ( A. z [ z / y ] ( y e. A -> ps ) <-> A. z ( z e. A -> [ z / y ] ps ) )
7		nfv 1551	. . . . 5 |- F/ z ( y e. A -> ps )
8	7	sb8 1879	. . . 4 |- ( A. y ( y e. A -> ps ) <-> A. z [ z / y ] ( y e. A -> ps ) )
9		df-ral 2489	. . . 4 |- ( A. z e. A [ z / y ] ps <-> A. z ( z e. A -> [ z / y ] ps ) )
10	6, 8, 9	3bitr4i 212	. . 3 |- ( A. y ( y e. A -> ps ) <-> A. z e. A [ z / y ] ps )
11		nfv 1551	. . . 4 |- F/ z ph
12		nfraldya.3	. . . 4 |- ( ph -> F/_ x A )
13		nfraldya.2	. . . . 5 |- F/ y ph
14		nfraldya.4	. . . . 5 |- ( ph -> F/ x ps )
15	13, 14	nfsbd 2005	. . . 4 |- ( ph -> F/ x [ z / y ] ps )
16	11, 12, 15	nfraldxy 2539	. . 3 |- ( ph -> F/ x A. z e. A [ z / y ] ps )
17	10, 16	nfxfrd 1498	. 2 |- ( ph -> F/ x A. y ( y e. A -> ps ) )
18	1, 17	nfxfrd 1498	1 |- ( ph -> F/ x A. y e. A ps )

Syntax hints:   -> wi 4   A.wal 1371   F/wnf 1483   [wsb 1785   e. wcel 2176   F/_wnfc 2335   A.wral 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  nfralya  2546