Higher-Order Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HOLE Home  >  Th. List  >  dfov2 GIF version

Theorem dfov2 75
 Description: Reverse direction of df-ov 73. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
dfov1.1 F:(α → (β → ∗))
dfov1.2 A:α
dfov1.3 B:β
dfov2.4 R⊧((FA)B)
Assertion
Ref Expression
dfov2 R⊧[AFB]

Proof of Theorem dfov2
StepHypRef Expression
1 dfov1.1 . . 3 F:(α → (β → ∗))
2 dfov1.2 . . 3 A:α
3 dfov1.3 . . 3 B:β
41, 2, 3wov 72 . 2 [AFB]:∗
5 dfov2.4 . 2 R⊧((FA)B)
65ax-cb1 29 . . 3 R:∗
71, 2, 3df-ov 73 . . 3 ⊤⊧(( = [AFB])((FA)B))
86, 7a1i 28 . 2 R⊧(( = [AFB])((FA)B))
94, 5, 8mpbirx 53 1 R⊧[AFB]
 Colors of variables: type var term Syntax hints:   → ht 2  ∗hb 3  kc 5   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  eqcomi  79  eqid  83  ded  84  ceq12  88  leq  91  beta  92  distrc  93  distrl  94  eqtri  95  oveq123  98  hbov  111  ovl  117
 Copyright terms: Public domain W3C validator