Theorem rals-no-surprise 17056

Description: Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of A that satisfy the antecedent. This is the restricted counterpart of als-no-surprise 17055, and follows from it by dfrals2 17038. Note that this holds without any assumption that A is inhabited; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails when nothing in A satisfies ph. (Contributed by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
rals-no-surprise	|- -. ( A.E. x e. A ( ph -> ps ) /\ A.E. x e. A ( ph -> -. ps ) )

Proof of Theorem rals-no-surprise

Step	Hyp	Ref	Expression
1		als-no-surprise 17055	. 2 |- -. ( A.E. x ( ( x e. A /\ ph ) -> ps ) /\ A.E. x ( ( x e. A /\ ph ) -> -. ps ) )
2		dfrals2 17038	. . 3 |- ( A.E. x e. A ( ph -> ps ) <-> A.E. x ( ( x e. A /\ ph ) -> ps ) )
3		dfrals2 17038	. . 3 |- ( A.E. x e. A ( ph -> -. ps ) <-> A.E. x ( ( x e. A /\ ph ) -> -. ps ) )
4	2, 3	anbi12i 464	. 2 |- ( ( A.E. x e. A ( ph -> ps ) /\ A.E. x e. A ( ph -> -. ps ) ) <-> ( A.E. x ( ( x e. A /\ ph ) -> ps ) /\ A.E. x ( ( x e. A /\ ph ) -> -. ps ) ) )
5	1, 4	mtbir 682	1 |- -. ( A.E. x e. A ( ph -> ps ) /\ A.E. x e. A ( ph -> -. ps ) )

Syntax hints:   -. wn 3   /\ wa 104   e. wcel 2209   A.E.wals 17034   A.E.wrals 17035

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-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-ral 2533  df-rex 2534  df-als 17036  df-rals 17037

This theorem is referenced by: (None)