Theorem exlimim 34625

Description: Closed form of exlimimd 34626. (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 2155	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		nfv 1915	. . 3 ⊢ Ⅎ𝑥𝜓
3		sp 2182	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	1, 2, 3	exlimd 2218	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))
5	4	impcom 410	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)