MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.36v Structured version   Visualization version   GIF version

Theorem 19.36v 2072
Description: Version of 19.36 2254 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
Assertion
Ref Expression
19.36v (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36v
StepHypRef Expression
1 19.35 1957 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 2065 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 325 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  19.36ivOLD  2073  19.12vvv  2075  19.12vv  2342  ax13lem2  2451  axext2  2752  vtocl2  3413  vtocl3  3414  bnj1090  31386  bj-spimvwt  32994  bj-spcimdv  33214  bj-spcimdvv  33215  19.36vv  39109
  Copyright terms: Public domain W3C validator