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Theorem hbex 2154
 Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2152, hbex 2154. (Revised by Wolf Lammen, 16-Oct-2021.)
Hypothesis
Ref Expression
hbex.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbex (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Proof of Theorem hbex
StepHypRef Expression
1 hbex.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nf5i 2022 . . 3 𝑥𝜑
32nfex 2152 . 2 𝑥𝑦𝜑
43nf5ri 2063 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479  ∃wex 1702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045 This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708 This theorem is referenced by: (None)
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