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Theorem hbex 2319
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 2317, hbex 2319. (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 2178 . . 3 𝑥𝜑
32nfex 2317 . 2 𝑥𝑦𝜑
43nf5ri 2218 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-or 827  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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