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Theorem bj-ax5ea 31666
Description: If a formula holds for some value of a variable not occurring in it, then it holds for all values of that variable. (Contributed by BJ, 28-Dec-2020.)
Assertion
Ref Expression
bj-ax5ea (∃𝑥𝜑 → ∀𝑥𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem bj-ax5ea
StepHypRef Expression
1 ax5e 1795 . 2 (∃𝑥𝜑𝜑)
2 ax-5 1793 . 2 (𝜑 → ∀𝑥𝜑)
31, 2syl 17 1 (∃𝑥𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1793
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  bj-nfv  31667
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