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Theorem axi12 2429
 Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2140 (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 2150 . . . . 5
2 nfae 2150 . . . . 5
31, 2nfor 2018 . . . 4
4319.32 2047 . . 3
5 axc9 2140 . . . . . 6
65orrd 380 . . . . 5
76orri 378 . . . 4
8 orass 527 . . . 4
97, 8mpbir 213 . . 3
104, 9mpgbi 1672 . 2
11 orass 527 . 2
1210, 11mpbi 212 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 370  wal 1442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668 This theorem is referenced by: (None)
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