Theorem hba1 1786

Description: x is not free in ∀xφ. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.)

Assertion

Ref	Expression
hba1	⊢ (∀xφ → ∀x∀xφ)

Proof of Theorem hba1

Step	Hyp	Ref	Expression
1		hbe1 1731	. . 3 ⊢ (∃x ¬ φ → ∀x∃x ¬ φ)
2	1	hbn 1776	. 2 ⊢ (¬ ∃x ¬ φ → ∀x ¬ ∃x ¬ φ)
3		alex 1572	. 2 ⊢ (∀xφ ↔ ¬ ∃x ¬ φ)
4	3	albii 1566	. 2 ⊢ (∀x∀xφ ↔ ∀x ¬ ∃x ¬ φ)
5	2, 3, 4	3imtr4i 257	1 ⊢ (∀xφ → ∀x∀xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

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

This theorem is referenced by:  nfa1  1788  spimehOLD  1821  19.21hOLD  1840  19.12OLD  1848  cbv3hvOLD  1851  nfald  1852  ax12olem5  1931  ax10lem4  1941  ax9  1949  dvelimh  1964  axi5r  2326  axial  2327  hbra1  2664