Theorem exellim 37915

Description: Closed form of exellimddv 37916. See also exlimim 37913 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 2192	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑)
2		nfv 1941	. . 3 ⊢ Ⅎ𝑥𝜑
3		sp 2225	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))
4	1, 2, 3	exlimd 2260	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
5	4	impcom 412	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811

This theorem is referenced by:  exellimddv  37916