Theorem bj-ax5ea 31805

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

Step	Hyp	Ref	Expression
1		ax5e 1829	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		ax-5 1827	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	syl 17	1 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:   → 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-5 1827

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  bj-nfv  31806