Theorem 19.9 1637

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 1512	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.9h 1636	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  Ⅎwnf 1453  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  alexim  1638  19.19  1659  19.36-1  1666  19.44  1675  19.45  1676  19.41  1679