Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axi12 Unicode version

Theorem axi12 2433
 Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2046 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.)
Assertion
Ref Expression
axi12

Proof of Theorem axi12
StepHypRef Expression
1 nfae 2056 . . . . 5
2 nfae 2056 . . . . 5
31, 2nfor 1935 . . . 4
4319.32 1967 . . 3
5 axc9 2046 . . . . . 6
65orrd 378 . . . . 5
76orri 376 . . . 4
8 orass 524 . . . 4
97, 8mpbir 209 . . 3
104, 9mpgbi 1621 . 2
11 orass 524 . 2
1210, 11mpbi 208 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  \/wo 368  A.wal 1393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617
 Copyright terms: Public domain W3C validator