Theorem alsi-no-surprise 42542

Description: Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-alsi 42534; the proof itself builds on alimp-no-surprise 42527. For a contrast, see alimp-surprise 42526. (Contributed by David A. Wheeler, 27-Oct-2018.)

Assertion

Ref	Expression
alsi-no-surprise	|- -. ( A.! x ( ph -> ps ) /\ A.! x ( ph -> -. ps ) )

Proof of Theorem alsi-no-surprise

Step	Hyp	Ref	Expression
1		alimp-no-surprise 42527	. 2 |- -. ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph )
2		df-alsi 42534	. . . 4 |- ( A.! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
3		df-alsi 42534	. . . 4 |- ( A.! x ( ph -> -. ps ) <-> ( A. x ( ph -> -. ps ) /\ E. x ph ) )
4	2, 3	anbi12i 733	. . 3 |- ( ( A.! x ( ph -> ps ) /\ A.! x ( ph -> -. ps ) ) <-> ( ( A. x ( ph -> ps ) /\ E. x ph ) /\ ( A. x ( ph -> -. ps ) /\ E. x ph ) ) )
5		anandi3r 1053	. . 3 |- ( ( A. x ( ph -> ps ) /\ E. x ph /\ A. x ( ph -> -. ps ) ) <-> ( ( A. x ( ph -> ps ) /\ E. x ph ) /\ ( A. x ( ph -> -. ps ) /\ E. x ph ) ) )
6		3ancomb 1047	. . 3 |- ( ( A. x ( ph -> ps ) /\ E. x ph /\ A. x ( ph -> -. ps ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph ) )
7	4, 5, 6	3bitr2i 288	. 2 |- ( ( A.! x ( ph -> ps ) /\ A.! x ( ph -> -. ps ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph ) )
8	1, 7	mtbir 313	1 |- -. ( A.! x ( ph -> ps ) /\ A.! x ( ph -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   E.wex 1704   A.!walsi 42532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-alsi 42534

This theorem is referenced by: (None)