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Theorem 19.9 2060
 Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1883 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 2059 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  exlimd  2074  19.19  2084  19.36  2085  19.41  2090  19.44  2093  19.45  2094  19.9h  2106  exists1  2549  dfid3  4954  fsplit  7169  bnj1189  30331  bj-exexbiex  31878  bj-exalbial  31880  ax6e2ndeq  37796  e2ebind  37800  ax6e2ndeqVD  38167  e2ebindVD  38170  e2ebindALT  38187  ax6e2ndeqALT  38189
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