Theorem nfrexdxy 2540

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2542 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfrexdxy

Step	Hyp	Ref	Expression
1		df-rex 2490	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		nfraldxy.2	. . 3 |- F/ y ph
3		nfcv 2348	. . . . . 6 |- F/_ x y
4	3	a1i 9	. . . . 5 |- ( ph -> F/_ x y )
5		nfraldxy.3	. . . . 5 |- ( ph -> F/_ x A )
6	4, 5	nfeld 2364	. . . 4 |- ( ph -> F/ x y e. A )
7		nfraldxy.4	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfand 1591	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
9	2, 8	nfexd 1784	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
10	1, 9	nfxfrd 1498	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1483   E.wex 1515   e. wcel 2176   F/_wnfc 2335   E.wrex 2485

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  nfrexdya  2542  nfrexw  2545  nfunid  3857  strcollnft  15957