Theorem 19.9 2198

Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2195 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1987 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 2197	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  exlimd  2211  19.19  2222  19.36  2223  19.41  2228  19.44  2230  19.45  2231  19.9h  2283  eeor  2330  dfid3  5492  fsplitOLD  7958  bnj1189  32989  bj-exexbiex  34882  bj-exalbial  34884  ax6e2ndeq  42179  e2ebind  42183  ax6e2ndeqVD  42529  e2ebindVD  42532  e2ebindALT  42549  ax6e2ndeqALT  42551