Theorem nfralya 2537

Description: Not-free for restricted universal quantification where y and A are distinct. See nfralxy 2535 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 1480	. . 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 2532	. 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 1474   F/_wnfc 2326   A.wral 2475

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  nfiinya  3945  nfsup  7056  caucvgsrlemgt1  7860  axpre-suploclemres  7966  supinfneg  9666  infsupneg  9667  ctiunctlemudc  12630  trirec0  15655