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Theorem 19.23 2196
Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1937 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 2195 . 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 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  exlimi  2202  equsalv  2250  nf5  2270  19.23h  2276  pm11.53  2334  equsal  2408  2sb6rf  2464  ceqsal  3502  r19.3rz  4489  ralidmOLD  4508  ssrelf  32338  bj-biexal1  36083  bj-biexex  36087  axc11n-16  38311  axc11next  43714  r19.3rzf  44400
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