Theorem exlimim 38470

Description: Closed form of exlimimd 38471. (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 2191	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		nfv 1949	. . 3 ⊢ Ⅎ𝑥𝜓
3		sp 2223	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4	1, 2, 3	exlimd 2258	. 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 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by: (None)