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

Theorem hbral 2462
 Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
Hypotheses
Ref Expression
hbral.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
hbral.2 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbral (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)

Proof of Theorem hbral
StepHypRef Expression
1 df-ral 2419 . 2 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
2 hbral.1 . . . 4 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbral.2 . . . 4 (𝜑 → ∀𝑥𝜑)
42, 3hbim 1524 . . 3 ((𝑦𝐴𝜑) → ∀𝑥(𝑦𝐴𝜑))
54hbal 1453 . 2 (∀𝑦(𝑦𝐴𝜑) → ∀𝑥𝑦(𝑦𝐴𝜑))
61, 5hbxfrbi 1448 1 (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329   ∈ wcel 1480  ∀wral 2414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-4 1487  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-ral 2419 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator