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Theorem 19.23 2083
Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1904 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypothesis
Ref Expression
19.23.1 𝑥𝜓
Assertion
Ref Expression
19.23 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.23
StepHypRef Expression
1 19.23.1 . 2 𝑥𝜓
2 19.23t 2082 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  exlimi  2089  nf5  2118  19.23h  2124  pm11.53  2183  equsal  2295  2sb6rf  2456  r19.3rz  4039  ralidm  4052  ssrelf  29259  bj-biexal1  32330  bj-biexex  32334  bj-equsalv  32378  axc11n-16  33689  axc11next  38075
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