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Theorem hbral 2927
 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 2901 . 2 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
2 hbral.1 . . . 4 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbral.2 . . . 4 (𝜑 → ∀𝑥𝜑)
42, 3hbim 2112 . . 3 ((𝑦𝐴𝜑) → ∀𝑥(𝑦𝐴𝜑))
54hbal 2023 . 2 (∀𝑦(𝑦𝐴𝜑) → ∀𝑥𝑦(𝑦𝐴𝜑))
61, 5hbxfrbi 1742 1 (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-ral 2901 This theorem is referenced by:  tratrbVD  38119
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