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Theorem hbnd 2143
Description: Deduction form of bound-variable hypothesis builder hbn 2142. (Contributed by NM, 3-Jan-2002.)
Hypotheses
Ref Expression
hbnd.1 (𝜑 → ∀𝑥𝜑)
hbnd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
hbnd (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Proof of Theorem hbnd
StepHypRef Expression
1 hbnd.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 hbnd.2 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
31, 2alrimih 1748 . 2 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
4 hbnt 2140 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
53, 4syl 17 1 (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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