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 2136
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1960 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 1845 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36.1 . . . 4 𝑥𝜓
3219.9 2110 . . 3 (∃𝑥𝜓𝜓)
43imbi2i 325 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by:  19.36i  2137  19.12vv  2216  spcimgft  3315  19.12b  31831
  Copyright terms: Public domain W3C validator