Theorem 19.9 1783

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	⊢ Ⅎxφ

Assertion

Ref	Expression
19.9	⊢ (∃xφ ↔ φ)

Proof of Theorem 19.9

Step	Hyp	Ref	Expression
1		19.9.1	. . 3 ⊢ Ⅎxφ
2	1	nfri 1762	. 2 ⊢ (φ → ∀xφ)
3	2	19.9h 1780	1 ⊢ (∃xφ ↔ φ)

Syntax hints:   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  excomimOLD  1858  19.19  1862  19.36  1871  19.44  1877  19.45  1878  exdistrf  1971  exists1  2293  dfid3  4768