Theorem nfrals 17053

Description: Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
nfrals.1	|- F/_ x A
nfrals.2	|- F/ x ph
nfrals.3	|- F/ x ps

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfrals

Step	Hyp	Ref	Expression
1		df-rals 17037	. 2 |- ( A.E. y e. A ( ph -> ps ) <-> ( A. y e. A ( ph -> ps ) /\ E. y e. A ph ) )
2		nfrals.1	. . . 4 |- F/_ x A
3		nfrals.2	. . . . 5 |- F/ x ph
4		nfrals.3	. . . . 5 |- F/ x ps
5	3, 4	nfim 1625	. . . 4 |- F/ x ( ph -> ps )
6	2, 5	nfralw 2587	. . 3 |- F/ x A. y e. A ( ph -> ps )
7	2, 3	nfrexw 2589	. . 3 |- F/ x E. y e. A ph
8	6, 7	nfan 1618	. 2 |- F/ x ( A. y e. A ( ph -> ps ) /\ E. y e. A ph )
9	1, 8	nfxfr 1527	1 |- F/ x A.E. y e. A ( ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1513   F/_wnfc 2379   A.wral 2528   E.wrex 2529   A.E.wrals 17035

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  df-rex 2534  df-rals 17037

This theorem is referenced by: (None)