Theorem 19.9 2247

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2244 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2011 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 2246	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  exlimd  2260  19.19  2271  19.36  2272  19.41  2277  19.44  2279  19.45  2280  19.9h  2327  eeor  2372  dfid3  5560  bnj1189  35341  bj-exexbiex  37213  bj-exalbial  37215  ax6e2ndeq  45159  e2ebind  45163  ax6e2ndeqVD  45508  e2ebindVD  45511  e2ebindALT  45528  ax6e2ndeqALT  45530