Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm10.55 Structured version   Visualization version   GIF version

Theorem pm10.55 42963
Description: Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.55 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))

Proof of Theorem pm10.55
StepHypRef Expression
1 exsimpl 1871 . . 3 (∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
21anim1i 615 . 2 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
3 exintr 1895 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
43imdistanri 570 . 2 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)))
52, 4impbii 208 1 ((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator