Theorem nfreuxy 2541

Description: Not-free for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)

Hypotheses

Ref	Expression
nfreuxy.1	|- F/_ x A
nfreuxy.2	|- F/ x ph

Assertion

Ref	Expression
nfreuxy	|- F/ x E! y e. A ph

Distinct variable group:   x, y

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

Proof of Theorem nfreuxy

Step	Hyp	Ref	Expression
1		nftru 1400	. . 3 |- F/ y T.
2		nfreuxy.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfreuxy.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfreudxy 2540	. 2 |- ( T. -> F/ x E! y e. A ph )
7	6	mptru 1298	1 |- F/ x E! y e. A ph

Syntax hints:   T. wtru 1290   F/wnf 1394   F/_wnfc 2215   E!wreu 2361

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366

This theorem is referenced by:  sbcreug  2919