Theorem nfrexdya 2401

Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2399 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 2354	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		sban 1870	. . . . . 6 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( [ z / y ] y e. A /\ [ z / y ] ps ) )
3		clelsb3 2183	. . . . . . 7 |- ( [ z / y ] y e. A <-> z e. A )
4	3	anbi1i 445	. . . . . 6 |- ( ( [ z / y ] y e. A /\ [ z / y ] ps ) <-> ( z e. A /\ [ z / y ] ps ) )
5	2, 4	bitri 182	. . . . 5 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( z e. A /\ [ z / y ] ps ) )
6	5	exbii 1536	. . . 4 |- ( E. z [ z / y ] ( y e. A /\ ps ) <-> E. z ( z e. A /\ [ z / y ] ps ) )
7		nfv 1461	. . . . 5 |- F/ z ( y e. A /\ ps )
8	7	sb8e 1778	. . . 4 |- ( E. y ( y e. A /\ ps ) <-> E. z [ z / y ] ( y e. A /\ ps ) )
9		df-rex 2354	. . . 4 |- ( E. z e. A [ z / y ] ps <-> E. z ( z e. A /\ [ z / y ] ps ) )
10	6, 8, 9	3bitr4i 210	. . 3 |- ( E. y ( y e. A /\ ps ) <-> E. z e. A [ z / y ] ps )
11		nfv 1461	. . . 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 1892	. . . 4 |- ( ph -> F/ x [ z / y ] ps )
16	11, 12, 15	nfrexdxy 2399	. . 3 |- ( ph -> F/ x E. z e. A [ z / y ] ps )
17	10, 16	nfxfrd 1404	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
18	1, 17	nfxfrd 1404	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1389   E.wex 1421   e. wcel 1433   [wsb 1685   F/_wnfc 2206   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by:  nfrexya  2405