Theorem exlimih 1804

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-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	nfi 1551	. 2 ⊢ Ⅎxψ
3		exlimih.2	. 2 ⊢ (φ → ψ)
4	2, 3	exlimi 1803	1 ⊢ (∃xφ → ψ)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is used by:  ax12olem5  1931  ax10lem2  1937  a16g  1945