Theorem 19.9 2240

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2237 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2004 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 2239	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1799  Ⅎwnf 1803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-ex 1800  df-nf 1804

This theorem is referenced by:  exlimd  2253  19.19  2264  19.36  2265  19.41  2270  19.44  2272  19.45  2273  19.9h  2320  eeor  2365  dfid3  5545  bnj1189  35304  bj-exexbiex  37175  bj-exalbial  37177  ax6e2ndeq  45135  e2ebind  45139  ax6e2ndeqVD  45484  e2ebindVD  45487  e2ebindALT  45504  ax6e2ndeqALT  45506