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Theorem 19.9 2023
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1844 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 2022 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 191  wex 1692  wnf 1696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-ex 1693  df-nf 1697
This theorem is referenced by:  19.9h  2024  exlimd  2050  19.19  2092  19.36  2096  19.44  2101  19.45  2102  19.41  2103  exists1  2444  dfid3  4796  fsplit  6980  bnj1189  29970  bj-exexbiex  31479  bj-exalbial  31481  ax6e2ndeq  37282  e2ebind  37286  ax6e2ndeqVD  37654  e2ebindVD  37657  e2ebindALT  37674  ax6e2ndeqALT  37676
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