Theorem axie2 2329

Description: A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axie2	⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Proof of Theorem axie2

Step	Hyp	Ref	Expression
1		df-nf 1545	. 2 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
2		19.23t 1800	. 2 ⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ)))
3	1, 2	sylbir 204	1 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)