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Theorem 19.23 2178
Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1924 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 2177 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1523  wex 1765  wnf 1769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-12 2143
This theorem depends on definitions:  df-bi 208  df-ex 1766  df-nf 1770
This theorem is referenced by:  exlimi  2184  equsalv  2233  nf5  2258  19.23h  2264  pm11.53  2325  equsal  2397  2sb6rf  2456  2sb6rfOLD  2457  r19.3rz  4362  ralidm  4375  ssrelf  30052  bj-biexal1  33643  bj-biexex  33647  axc11n-16  35626  axc11next  40297
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