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Theorem 19.23 2067
 Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1889 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 2066 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃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-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  exlimi  2073  nf5  2102  19.23h  2108  pm11.53  2167  equsal  2279  2sb6rf  2440  r19.3rz  4014  ralidm  4027  ssrelf  28805  bj-biexal1  31883  bj-biexex  31887  bj-equsalv  31931  axc11n-16  33241  axc11next  37629
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