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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)
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