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Theorem hbald 2025
Description: Deduction form of bound-variable hypothesis builder hbal 2021. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
hbald.1 (𝜑 → ∀𝑦𝜑)
hbald.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
hbald (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))

Proof of Theorem hbald
StepHypRef Expression
1 hbald.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 hbald.2 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
31, 2alimdh 1733 . 2 (𝜑 → (∀𝑦𝜓 → ∀𝑦𝑥𝜓))
4 ax-11 2019 . 2 (∀𝑦𝑥𝜓 → ∀𝑥𝑦𝜓)
53, 4syl6 34 1 (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1711  ax-4 1726  ax-11 2019
This theorem is referenced by:  nfald  2146  dvelimf-o  33031
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