Theorem nfralya 2572

Description: Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2570 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
nfralya	|- F/ x A. y e. A ph

Distinct variable group:   y, A

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

Proof of Theorem nfralya

Step	Hyp	Ref	Expression
1		nftru 1514	. . 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	nfraldya 2567	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1406	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1398   F/wnf 1508   F/_wnfc 2361   A.wral 2510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515

This theorem is referenced by:  nfiinya  3999  nfsup  7190  caucvgsrlemgt1  8014  axpre-suploclemres  8120  supinfneg  9828  infsupneg  9829  ctiunctlemudc  13057  trirec0  16648