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Axiom ax-ie2 1399
Description: Define existential quantification.  E. x ph means "there exists at least one set  x such that  ph is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ax-ie2  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x (
ph  ->  ps )  <->  ( E. x ph  ->  ps )
) )

Detailed syntax breakdown of Axiom ax-ie2
StepHypRef Expression
1 wps . . . 4  wff  ps
2 vx . . . . 5  setvar  x
31, 2wal 1257 . . . 4  wff  A. x ps
41, 3wi 4 . . 3  wff  ( ps 
->  A. x ps )
54, 2wal 1257 . 2  wff  A. x
( ps  ->  A. x ps )
6 wph . . . . 5  wff  ph
76, 1wi 4 . . . 4  wff  ( ph  ->  ps )
87, 2wal 1257 . . 3  wff  A. x
( ph  ->  ps )
96, 2wex 1397 . . . 4  wff  E. x ph
109, 1wi 4 . . 3  wff  ( E. x ph  ->  ps )
118, 10wb 102 . 2  wff  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )
125, 11wi 4 1  wff  ( A. x ( ps  ->  A. x ps )  -> 
( A. x (
ph  ->  ps )  <->  ( E. x ph  ->  ps )
) )
Colors of variables: wff set class
This axiom is referenced by:  19.23ht  1402  bj-ex  10268
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