Theorem 19.9 2212

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2209 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1985 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 2211	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785

This theorem is referenced by:  exlimd  2225  19.19  2236  19.36  2237  19.41  2242  19.44  2244  19.45  2245  19.9h  2292  eeor  2338  dfid3  5522  bnj1189  35165  bj-exexbiex  36901  bj-exalbial  36903  ax6e2ndeq  44800  e2ebind  44804  ax6e2ndeqVD  45149  e2ebindVD  45152  e2ebindALT  45169  ax6e2ndeqALT  45171