Theorem nfraldw 2502

Description: Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2505 for a version with y and A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfraldw.1	|- F/ y ph
nfraldw.2	|- ( ph -> F/_ x A )
nfraldw.3	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfraldw	|- ( ph -> F/ x A. y e. A ps )

Distinct variable group:   x, y

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

Proof of Theorem nfraldw

Step	Hyp	Ref	Expression
1		df-ral 2453	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
2		nfraldw.1	. . 3 |- F/ y ph
3		nfcvd 2313	. . . . 5 |- ( ph -> F/_ x y )
4		nfraldw.2	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeld 2328	. . . 4 |- ( ph -> F/ x y e. A )
6		nfraldw.3	. . . 4 |- ( ph -> F/ x ps )
7	5, 6	nfimd 1578	. . 3 |- ( ph -> F/ x ( y e. A -> ps ) )
8	2, 7	nfald 1753	. 2 |- ( ph -> F/ x A. y ( y e. A -> ps ) )
9	1, 8	nfxfrd 1468	1 |- ( ph -> F/ x A. y e. A ps )

Syntax hints:   -> wi 4   A.wal 1346   F/wnf 1453   e. wcel 2141   F/_wnfc 2299   A.wral 2448

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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  nfralw  2507