Theorem nfrexya 2547

Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexw 2545 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)

Hypotheses

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

Assertion

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

Distinct variable group:   y, A

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

Proof of Theorem nfrexya

Step	Hyp	Ref	Expression
1		nftru 1489	. . 3 |- F/ y T.
2		nfralya.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfralya.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfrexdya 2542	. 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 1483   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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  nfiunya  3955  nffrec  6482  nfsup  7094  caucvgsrlemgt1  7908  zsupcllemstep  10372  nfsum1  11667  bezout  12332