Theorem 19.9h 1780

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.) (Proof shortened by Wolf Lammen, 5-Jan-2018.)

Hypothesis

Ref	Expression
19.9h.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.9h	⊢ (∃xφ ↔ φ)

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9h.1	. . 3 ⊢ (φ → ∀xφ)
2	1	nfi 1551	. 2 ⊢ Ⅎxφ
3		19.9t 1779	. 2 ⊢ (Ⅎxφ → (∃xφ ↔ φ))
4	2, 3	ax-mp 5	1 ⊢ (∃xφ ↔ φ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃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:  19.9  1783  19.23hOLD  1820  cbv3hv  1850