Theorem nfralw 2502

Description: Bound-variable hypothesis builder for restricted quantification. See nfralya 2505 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfralw.1	|- F/_ x A
nfralw.2	|- F/ x ph

Assertion

Ref	Expression
nfralw	|- F/ x A. y e. A ph

Distinct variable group:   x, y

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

Proof of Theorem nfralw

Step	Hyp	Ref	Expression
1		nftru 1454	. . 3 |- F/ y T.
2		nfralw.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfralw.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfraldw 2497	. 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 2294   A.wral 2443

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448

This theorem is referenced by:  fprod2dlemstep  11559  fprodcom2fi  11563  nnwofdc  11967