Theorem 19.9 2228

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2065 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)

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	. 2 ⊢ Ⅎ𝑥𝜑
2		19.9t 2227	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1852  Ⅎwnf 1856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203

This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858

This theorem is referenced by:  exlimd  2243  19.19  2253  19.36  2254  19.41  2259  19.44  2262  19.45  2263  19.9h  2283  exists1  2710  dfid3  5159  fsplit  7434  bnj1189  31416  bj-exexbiex  33029  bj-exalbial  33031  ax6e2ndeq  39301  e2ebind  39305  ax6e2ndeqVD  39668  e2ebindVD  39671  e2ebindALT  39688  ax6e2ndeqALT  39690