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Theorem hbnt 1559
 Description: Closed theorem version of bound-variable hypothesis builder hbn 1560. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
Assertion
Ref Expression
hbnt (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbnt
StepHypRef Expression
1 ax-4 1416 . . . 4 (∀𝑥𝜑𝜑)
21con3i 572 . . 3 𝜑 → ¬ ∀𝑥𝜑)
3 ax6b 1557 . . 3 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
42, 3syl 14 . 2 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5 con3 581 . . 3 ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
65al2imi 1363 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
74, 6syl5 32 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265 This theorem is referenced by:  hbn  1560  hbnd  1561  nfnt  1562
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