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Theorem hbral 2353
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 2311 . 2 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
2 hbral.1 . . . 4 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbral.2 . . . 4 (𝜑 → ∀𝑥𝜑)
42, 3hbim 1437 . . 3 ((𝑦𝐴𝜑) → ∀𝑥(𝑦𝐴𝜑))
54hbal 1366 . 2 (∀𝑦(𝑦𝐴𝜑) → ∀𝑥𝑦(𝑦𝐴𝜑))
61, 5hbxfrbi 1361 1 (∀𝑦𝐴 𝜑 → ∀𝑥𝑦𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241  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-7 1337  ax-gen 1338  ax-4 1400  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-ral 2311
This theorem is referenced by: (None)
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