Theorem 19.9h 1577

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 1575	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
2		19.9h.1	. . 3 |- ( ph -> A. x ph )
3	1, 2	mpg 1383	. 2 |- ( E. x ph -> ph )
4		19.8a 1525	. 2 |- ( ph -> E. x ph )
5	3, 4	impbii 124	1 |- ( E. x ph <-> ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1285   E.wex 1424

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.9  1578  excomim  1596  exdistrfor  1725  sbcof2  1735  ax11ev  1753  19.9v  1796  exists1  2041