Theorem nfralw 2587

Description: Bound-variable hypothesis builder for restricted quantification. See nfralya 2590 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by GG, 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 1519	. . 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 2582	. 2 |- ( T. -> F/ x A. y e. A ph )
7	6	mptru 1411	1 |- F/ x A. y e. A ph

Syntax hints:   T. wtru 1403   F/wnf 1513   F/_wnfc 2379   A.wral 2528

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533

This theorem is referenced by:  reu8nf  3133  rspc2vd  3216  opabfi  7237  reuccatpfxs1  11497  fprod2dlemstep  12367  fprodcom2fi  12371  nnwofdc  12793