Theorem nfals 17052

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

Hypotheses

Ref	Expression
nfals.1	|- F/ x ph
nfals.2	|- F/ x ps

Assertion

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

Proof of Theorem nfals

Step	Hyp	Ref	Expression
1		df-als 17036	. 2 |- ( A.E. y ( ph -> ps ) <-> ( A. y ( ph -> ps ) /\ E. y ph ) )
2		nfals.1	. . . . 5 |- F/ x ph
3		nfals.2	. . . . 5 |- F/ x ps
4	2, 3	nfim 1625	. . . 4 |- F/ x ( ph -> ps )
5	4	nfal 1629	. . 3 |- F/ x A. y ( ph -> ps )
6	2	nfex 1690	. . 3 |- F/ x E. y ph
7	5, 6	nfan 1618	. 2 |- F/ x ( A. y ( ph -> ps ) /\ E. y ph )
8	1, 7	nfxfr 1527	1 |- F/ x A.E. y ( ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   F/wnf 1513   E.wex 1545   A.E.wals 17034

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-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-als 17036

This theorem is referenced by: (None)