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Theorem 19.23 2200
Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1938 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 2199 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wex 1774  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779
This theorem is referenced by:  exlimi  2206  equsalv  2254  nf5  2272  19.23h  2278  pm11.53  2338  equsal  2412  2sb6rf  2468  ceqsal  3507  r19.3rz  4497  ralidmOLD  4516  ssrelf  32404  bj-biexal1  36182  bj-biexex  36186  axc11n-16  38410  axc11next  43843  r19.3rzf  44529
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