Theorem alseuals 17139

Description: "All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise 17153 is proved. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

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

Proof of Theorem alseuals

Step	Hyp	Ref	Expression
1		euex 2116	. . 3 |- ( E! x ph -> E. x ph )
2	1	anim2i 342	. 2 |- ( ( A. x ( ph -> ps ) /\ E! x ph ) -> ( A. x ( ph -> ps ) /\ E. x ph ) )
3		df-alseu 17136	. 2 |- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
4		df-als 17102	. 2 |- ( A.E. x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
5	2, 3, 4	3imtr4i 201	1 |- ( A.E! x ( ph -> ps ) -> A.E. x ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   E.wex 1545   E!weu 2086   A.E.wals 17100   A.E!walseu 17134

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-als 17102  df-alseu 17136

This theorem is referenced by:  alseu-no-surprise  17153