Theorem 19.9 2190

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2186 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2030 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 2189	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1823  Ⅎwnf 1827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828

This theorem is referenced by:  exlimd  2204  19.19  2215  19.36  2216  19.41  2221  19.44  2224  19.45  2225  19.9h  2260  euaeOLD  2691  dfid3  5264  fsplit  7565  bnj1189  31680  bj-exexbiex  33283  bj-exalbial  33285  ax6e2ndeq  39729  e2ebind  39733  ax6e2ndeqVD  40088  e2ebindVD  40091  e2ebindALT  40108  ax6e2ndeqALT  40110