New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  exanali GIF version

Theorem exanali 1585
 Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
Assertion
Ref Expression
exanali (x(φ ¬ ψ) ↔ ¬ x(φψ))

Proof of Theorem exanali
StepHypRef Expression
1 annim 414 . . 3 ((φ ¬ ψ) ↔ ¬ (φψ))
21exbii 1582 . 2 (x(φ ¬ ψ) ↔ x ¬ (φψ))
3 exnal 1574 . 2 (x ¬ (φψ) ↔ ¬ x(φψ))
42, 3bitri 240 1 (x(φ ¬ ψ) ↔ ¬ x(φψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  ax11indn  2195  rexnal  2625  gencbval  2903  nss  3329  ssfin  4470  ncfinlowerlem1  4482  spfinex  4537  nfunv  5138  funsex  5828  fnfullfunlem1  5856  foundex  5914  fnfreclem1  6317
 Copyright terms: Public domain W3C validator