Theorem exlimii 35014

Description: Inference associated with exlimi 2210. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.)

Hypotheses

Ref	Expression
exlimii.1	⊢ Ⅎ𝑥𝜓
exlimii.2	⊢ (𝜑 → 𝜓)
exlimii.3	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
exlimii	⊢ 𝜓

Proof of Theorem exlimii

Step	Hyp	Ref	Expression
1		exlimii.3	. 2 ⊢ ∃𝑥𝜑
2		exlimii.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		exlimii.2	. . 3 ⊢ (𝜑 → 𝜓)
4	2, 3	exlimi 2210	. 2 ⊢ (∃𝑥𝜑 → 𝜓)
5	1, 4	ax-mp 5	1 ⊢ 𝜓

Syntax hints:   → wi 4  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  exlimiieq1  35017  exlimiieq2  35018