Theorem exellim 32368

Description: Closed form of exellimddv 32369. See also exlimim 32365 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 2015	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑)
2		nfv 1830	. . 3 ⊢ Ⅎ𝑥𝜑
3		sp 2041	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))
4	1, 2, 3	exlimd 2074	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
5	4	impcom 445	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  exellimddv  32369