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Theorem hbnd 1545
 Description: Deduction form of bound-variable hypothesis builder hbn 1544. (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 1358 . 2 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
4 hbnt 1543 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
53, 4syl 14 1 (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241 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-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249 This theorem is referenced by: (None)
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