Theorem 19.9 2241

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2238 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2016 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 2240	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  exlimd  2254  19.19  2265  19.36  2266  19.41  2271  19.44  2273  19.45  2274  19.9h  2319  eeor  2363  dfid3  5549  bnj1189  35689  bj-exexbiex  37698  bj-exalbial  37700  ax6e2ndeq  45650  e2ebind  45654  ax6e2ndeqVD  45999  e2ebindVD  46002  e2ebindALT  46019  ax6e2ndeqALT  46021