Theorem nfralya 2506

Description: Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2504 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 1454	. . 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 2501	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1352	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1344   F/wnf 1448   F/_wnfc 2295   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  nfiinya  3895  nfsup  6957  caucvgsrlemgt1  7736  axpre-suploclemres  7842  supinfneg  9533  infsupneg  9534  ctiunctlemudc  12370  trirec0  13923