Theorem nfrexdya 2359

Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2357 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
nfrexdya	|- ( ph -> F/ x E. y e. A ps )

Distinct variable group:   y, A

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

Proof of Theorem nfrexdya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2312	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		sban 1829	. . . . . 6 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( [ z / y ] y e. A /\ [ z / y ] ps ) )
3		clelsb3 2142	. . . . . . 7 |- ( [ z / y ] y e. A <-> z e. A )
4	3	anbi1i 431	. . . . . 6 |- ( ( [ z / y ] y e. A /\ [ z / y ] ps ) <-> ( z e. A /\ [ z / y ] ps ) )
5	2, 4	bitri 173	. . . . 5 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( z e. A /\ [ z / y ] ps ) )
6	5	exbii 1496	. . . 4 |- ( E. z [ z / y ] ( y e. A /\ ps ) <-> E. z ( z e. A /\ [ z / y ] ps ) )
7		nfv 1421	. . . . 5 |- F/ z ( y e. A /\ ps )
8	7	sb8e 1737	. . . 4 |- ( E. y ( y e. A /\ ps ) <-> E. z [ z / y ] ( y e. A /\ ps ) )
9		df-rex 2312	. . . 4 |- ( E. z e. A [ z / y ] ps <-> E. z ( z e. A /\ [ z / y ] ps ) )
10	6, 8, 9	3bitr4i 201	. . 3 |- ( E. y ( y e. A /\ ps ) <-> E. z e. A [ z / y ] ps )
11		nfv 1421	. . . 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 1851	. . . 4 |- ( ph -> F/ x [ z / y ] ps )
16	11, 12, 15	nfrexdxy 2357	. . 3 |- ( ph -> F/ x E. z e. A [ z / y ] ps )
17	10, 16	nfxfrd 1364	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
18	1, 17	nfxfrd 1364	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 97   F/wnf 1349   E.wex 1381   e. wcel 1393   [wsb 1645   F/_wnfc 2165   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312

This theorem is referenced by:  nfrexya  2363