Theorem exlimih 2326

Description: Inference associated with 19.23 2250. See exlimiv 1963 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypotheses

Ref	Expression
exlimih.1	⊢ (𝜓 → ∀𝑥𝜓)
exlimih.2	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
exlimih	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem exlimih

Step	Hyp	Ref	Expression
1		exlimih.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2184	. 2 ⊢ Ⅎ𝑥𝜓
3		exlimih.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	exlimi 2256	1 ⊢ (∃𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀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 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)