Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralm GIF version

Theorem ralm 3325
 Description: Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
ralm ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralm
StepHypRef Expression
1 df-ral 2311 . . . . . 6 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
21imbi2i 215 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
3 19.38 1566 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
42, 3sylbi 114 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
5 pm2.43 47 . . . . 5 ((𝑥𝐴 → (𝑥𝐴𝜑)) → (𝑥𝐴𝜑))
65alimi 1344 . . . 4 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴𝜑))
74, 6syl 14 . . 3 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴𝜑))
87, 1sylibr 137 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥𝐴 𝜑)
9 ax-1 5 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
108, 9impbii 117 1 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-ral 2311 This theorem is referenced by:  raaan  3327
 Copyright terms: Public domain W3C validator