Theorem nfralya 2530

Description: Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2528 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 1477	. . 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 2525	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1373	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1365   F/wnf 1471   F/_wnfc 2319   A.wral 2468

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473

This theorem is referenced by:  nfiinya  3930  nfsup  7021  caucvgsrlemgt1  7824  axpre-suploclemres  7930  supinfneg  9625  infsupneg  9626  ctiunctlemudc  12488  trirec0  15251