Theorem ralseurals 17140

Description: "All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 17139. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

Ref	Expression
ralseurals	|- ( A.E! x e. A ( ph -> ps ) -> A.E. x e. A ( ph -> ps ) )

Proof of Theorem ralseurals

Step	Hyp	Ref	Expression
1		reurex 2771	. . 3 |- ( E! x e. A ph -> E. x e. A ph )
2	1	anim2i 342	. 2 |- ( ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) -> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
3		df-ralseu 17137	. 2 |- ( A.E! x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) )
4		df-rals 17103	. 2 |- ( A.E. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
5	2, 3, 4	3imtr4i 201	1 |- ( A.E! x e. A ( ph -> ps ) -> A.E. x e. A ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2528   E.wrex 2529   E!wreu 2530   A.E.wrals 17101   A.E!wralseu 17135

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-rex 2534  df-reu 2535  df-rmo 2536  df-rals 17103  df-ralseu 17137

This theorem is referenced by: (None)