Theorem 19.9h 1590

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)

Hypothesis

Ref	Expression
19.9h.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
19.9h	|- ( E. x ph <-> ph )

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9ht 1588	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
2		19.9h.1	. . 3 |- ( ph -> A. x ph )
3	1, 2	mpg 1395	. 2 |- ( E. x ph -> ph )
4		19.8a 1537	. 2 |- ( ph -> E. x ph )
5	3, 4	impbii 125	1 |- ( E. x ph <-> ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1297   E.wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.9  1591  excomim  1609  exdistrfor  1739  sbcof2  1749  ax11ev  1767  19.9v  1810  exists1  2056