Theorem 19.9 2206

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2203 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1983 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 2205	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by:  exlimd  2219  19.19  2230  19.36  2231  19.41  2236  19.44  2238  19.45  2239  19.9h  2290  eeor  2339  dfid3  5596  bnj1189  34985  bj-exexbiex  36666  bj-exalbial  36668  ax6e2ndeq  44530  e2ebind  44534  ax6e2ndeqVD  44880  e2ebindVD  44883  e2ebindALT  44900  ax6e2ndeqALT  44902