Theorem alseueu 17152

Description: "The ph is ps " implies that exactly one thing is both ph and ps. This is the half of dfalseu2 17151 that drops the universal conjunct; it does not reverse, so E! x ( ph /\ ps ) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

Ref	Expression
alseueu	|- ( A.E! x ( ph -> ps ) -> E! x ( ph /\ ps ) )

Proof of Theorem alseueu

Step	Hyp	Ref	Expression
1		dfalseu2 17151	. 2 |- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ( ph /\ ps ) ) )
2	1	simprbi 275	1 |- ( A.E! x ( ph -> ps ) -> E! x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   E!weu 2086   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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-eu 2089  df-alseu 17136

This theorem is referenced by: (None)