Theorem exellim 37713

Description: Closed form of exellimddv 37714. See also exlimim 37711 for a more general theorem. (Contributed by ML, 17-Jul-2020.)

Assertion

Ref	Expression
exellim	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exellim

Step	Hyp	Ref	Expression
1		nfa1 2162	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑)
2		nfv 1921	. . 3 ⊢ Ⅎ𝑥𝜑
3		sp 2195	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))
4	1, 2, 3	exlimd 2230	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
5	4	impcom 408	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786   ∈ wcel 2119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  exellimddv  37714