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

Theorem 19.36 2222
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1985 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
Hypothesis
Ref Expression
19.36.1 𝑥𝜓
Assertion
Ref Expression
19.36 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.36
StepHypRef Expression
1 19.35 1869 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36.1 . . . 4 𝑥𝜓
3219.9 2195 . . 3 (∃𝑥𝜓𝜓)
43imbi2i 337 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 276 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by:  19.36i  2223  19.12vv  2359  spcimgft  3583  19.12b  32943
  Copyright terms: Public domain W3C validator