Theorem nfreuw 2683

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
nfreuw.1	|- F/_ x A
nfreuw.2	|- F/ x ph

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfreuw

Step	Hyp	Ref	Expression
1		nftru 1490	. . 3 |- F/ y T.
2		nfreuw.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfreuw.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfreudxy 2682	. 2 |- ( T. -> F/ x E! y e. A ph )
7	6	mptru 1382	1 |- F/ x E! y e. A ph

Syntax hints:   T. wtru 1374   F/wnf 1484   F/_wnfc 2337   E!wreu 2488

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-cleq 2200  df-clel 2203  df-nfc 2339  df-reu 2493

This theorem is referenced by:  sbcreug  3086  reuccatpfxs1  11238