Theorem 19.9 2205

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2202 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1993 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 2204	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792

This theorem is referenced by:  exlimd  2218  19.19  2229  19.36  2230  19.41  2235  19.44  2237  19.45  2238  19.9h  2289  dfid3  5442  fsplitOLD  7864  bnj1189  32656  bj-exexbiex  34568  bj-exalbial  34570  ax6e2ndeq  41793  e2ebind  41797  ax6e2ndeqVD  42143  e2ebindVD  42146  e2ebindALT  42163  ax6e2ndeqALT  42165