Theorem 19.9 2217

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2214 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1991 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 2216	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  exlimd  2230  19.19  2241  19.36  2242  19.41  2247  19.44  2249  19.45  2250  19.9h  2297  eeor  2342  dfid3  5516  bnj1189  35191  bj-exexbiex  37043  bj-exalbial  37045  ax6e2ndeq  45003  e2ebind  45007  ax6e2ndeqVD  45352  e2ebindVD  45355  e2ebindALT  45372  ax6e2ndeqALT  45374