Theorem 19.9 1690

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 1565	. 2 |- ( ph -> A. x ph )
3	2	19.9h 1689	1 |- ( E. x ph <-> ph )

Syntax hints:   <-> wb 105   F/wnf 1506   E.wex 1538

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 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  alexim  1691  19.19  1712  19.36-1  1719  19.44  1728  19.45  1729  19.41  1732