Theorem alsanmo 17059

Description: An "all some" statement conjoined with the claim that at most one x satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one x satisfies the antecedent. The "all some" quantifier supplies the existence of such an x and E* x ph supplies the at-most-one part, so together they yield E! x ph. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)

Assertion

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

Proof of Theorem alsanmo

Step	Hyp	Ref	Expression
1		df-als 17036	. . 3 |- ( A.E. x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
2	1	anbi1i 462	. 2 |- ( ( A.E. x ( ph -> ps ) /\ E* x ph ) <-> ( ( A. x ( ph -> ps ) /\ E. x ph ) /\ E* x ph ) )
3		anass 405	. 2 |- ( ( ( A. x ( ph -> ps ) /\ E. x ph ) /\ E* x ph ) <-> ( A. x ( ph -> ps ) /\ ( E. x ph /\ E* x ph ) ) )
4		eu5 2134	. . . 4 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
5	4	bicomi 132	. . 3 |- ( ( E. x ph /\ E* x ph ) <-> E! x ph )
6	5	anbi2i 461	. 2 |- ( ( A. x ( ph -> ps ) /\ ( E. x ph /\ E* x ph ) ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
7	2, 3, 6	3bitri 206	1 |- ( ( A.E. x ( ph -> ps ) /\ E* x ph ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   E.wex 1545   E!weu 2086   E*wmo 2087   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-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-als 17036

This theorem is referenced by: (None)