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Theorem 19.9 2204
 Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2201 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1988 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 𝑥𝜑
Assertion
Ref Expression
19.9 (∃𝑥𝜑𝜑)

Proof of Theorem 19.9
StepHypRef Expression
1 19.9.1 . 2 𝑥𝜑
2 19.9t 2203 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  exlimd  2217  19.19  2230  19.36  2231  19.41  2236  19.44  2238  19.45  2239  19.9h  2292  dfid3  5430  fsplitOLD  7800  bnj1189  32395  bj-exexbiex  34148  bj-exalbial  34150  ax6e2ndeq  41258  e2ebind  41262  ax6e2ndeqVD  41608  e2ebindVD  41611  e2ebindALT  41628  ax6e2ndeqALT  41630
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