MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbral Structured version   Visualization version   GIF version

Theorem hbral 2972
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 2946 . 2 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
2 hbral.1 . . . 4 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbral.2 . . . 4 (𝜑 → ∀𝑥𝜑)
42, 3hbim 2165 . . 3 ((𝑦𝐴𝜑) → ∀𝑥(𝑦𝐴𝜑))
54hbal 2076 . 2 (∀𝑦(𝑦𝐴𝜑) → ∀𝑥𝑦(𝑦𝐴𝜑))
61, 5hbxfrbi 1792 1 (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750  df-ral 2946
This theorem is referenced by:  tratrbVD  39411
  Copyright terms: Public domain W3C validator