Theorem 19.9 1655

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.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
19.9.1	|- F/ x ph

Assertion

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

Proof of Theorem 19.9

Step	Hyp	Ref	Expression
1		19.9.1	. . 3 |- F/ x ph
2	1	nfri 1530	. 2 |- ( ph -> A. x ph )
3	2	19.9h 1654	1 |- ( E. x ph <-> ph )

Syntax hints:   <-> wb 105   F/wnf 1471   E.wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  alexim  1656  19.19  1677  19.36-1  1684  19.44  1693  19.45  1694  19.41  1697