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Theorem hbexOLD 2138
 Description: Obsolete proof of hbex 2142 as of 16-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
hbexOLD.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbexOLD (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Proof of Theorem hbexOLD
StepHypRef Expression
1 df-ex 1696 . 2 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
2 hbexOLD.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
32hbn 2131 . . . 4 𝜑 → ∀𝑥 ¬ 𝜑)
43hbal 2023 . . 3 (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)
54hbn 2131 . 2 (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
61, 5hbxfrbi 1742 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  nfexOLD  2141
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