Theorem nfrexdya 2506

Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexdxy 2504 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 2454	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		sban 1948	. . . . . 6 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( [ z / y ] y e. A /\ [ z / y ] ps ) )
3		clelsb1 2275	. . . . . . 7 |- ( [ z / y ] y e. A <-> z e. A )
4	3	anbi1i 455	. . . . . 6 |- ( ( [ z / y ] y e. A /\ [ z / y ] ps ) <-> ( z e. A /\ [ z / y ] ps ) )
5	2, 4	bitri 183	. . . . 5 |- ( [ z / y ] ( y e. A /\ ps ) <-> ( z e. A /\ [ z / y ] ps ) )
6	5	exbii 1598	. . . 4 |- ( E. z [ z / y ] ( y e. A /\ ps ) <-> E. z ( z e. A /\ [ z / y ] ps ) )
7		nfv 1521	. . . . 5 |- F/ z ( y e. A /\ ps )
8	7	sb8e 1850	. . . 4 |- ( E. y ( y e. A /\ ps ) <-> E. z [ z / y ] ( y e. A /\ ps ) )
9		df-rex 2454	. . . 4 |- ( E. z e. A [ z / y ] ps <-> E. z ( z e. A /\ [ z / y ] ps ) )
10	6, 8, 9	3bitr4i 211	. . 3 |- ( E. y ( y e. A /\ ps ) <-> E. z e. A [ z / y ] ps )
11		nfv 1521	. . . 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 1970	. . . 4 |- ( ph -> F/ x [ z / y ] ps )
16	11, 12, 15	nfrexdxy 2504	. . 3 |- ( ph -> F/ x E. z e. A [ z / y ] ps )
17	10, 16	nfxfrd 1468	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
18	1, 17	nfxfrd 1468	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   F/wnf 1453   E.wex 1485   [wsb 1755   e. wcel 2141   F/_wnfc 2299   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  nfrexya  2511