Theorem eximp-surprise 42530

Description: Show what implication inside "there exists" really expands to (using implication directly inside "there exists" is usually a mistake).

Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. That is usually a mistake, because as proven using imor 428, such an expression can be rewritten using not with or - and that is often not what the author intended. New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". A stark example is shown in eximp-surprise2 42531. See also alimp-surprise 42526 and empty-surprise 42528. (Contributed by David A. Wheeler, 17-Oct-2018.)

Assertion

Ref	Expression
eximp-surprise	|- ( E. x ( ph -> ps ) <-> E. x ( -. ph \/ ps ) )

Proof of Theorem eximp-surprise

Step	Hyp	Ref	Expression
1		imor 428	. 2 |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
2	1	exbii 1774	1 |- ( E. x ( ph -> ps ) <-> E. x ( -. ph \/ ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   E.wex 1704

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-or 385  df-ex 1705

This theorem is referenced by:  eximp-surprise2  42531