Theorem 19.9 1591

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 1467	. 2 |- ( ph -> A. x ph )
3	2	19.9h 1590	1 |- ( E. x ph <-> ph )

Syntax hints:   <-> wb 104   F/wnf 1404   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  df-nf 1405

This theorem is referenced by:  alexim  1592  19.19  1612  19.36-1  1619  19.44  1628  19.45  1629  19.41  1632