Theorem exlimim 37284

Description: Closed form of exlimimd 37285. (Contributed by ML, 17-Jul-2020.)

Assertion

Ref	Expression
exlimim	⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exlimim

Step	Hyp	Ref	Expression
1		nfa1 2150	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		nfv 1913	. . 3 ⊢ Ⅎ𝑥𝜓
3		sp 2182	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	1, 2, 3	exlimd 2217	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))
5	4	impcom 407	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)