Theorem exlimim 38359

Description: Closed form of exlimimd 38360. (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 2187	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		nfv 1946	. . 3 ⊢ Ⅎ𝑥𝜓
3		sp 2219	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	1, 2, 3	exlimd 2254	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))
5	4	impcom 413	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by: (None)