Theorem hba1-o 2149

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.) (New usage is discouraged.)

Assertion

Ref	Expression
hba1-o	⊢ (∀xφ → ∀x∀xφ)

Proof of Theorem hba1-o

Step	Hyp	Ref	Expression
1		ax-4 2135	. . 3 ⊢ (∀x ¬ ∀xφ → ¬ ∀xφ)
2	1	con2i 112	. 2 ⊢ (∀xφ → ¬ ∀x ¬ ∀xφ)
3		ax6 2147	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ∀x ¬ ∀x ¬ ∀xφ)
4		ax6 2147	. . . 4 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
5	4	con1i 121	. . 3 ⊢ (¬ ∀x ¬ ∀xφ → ∀xφ)
6	5	alimi 1559	. 2 ⊢ (∀x ¬ ∀x ¬ ∀xφ → ∀x∀xφ)
7	2, 3, 6	3syl 18	1 ⊢ (∀xφ → ∀x∀xφ)

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

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-4 2135  ax-5o 2136  ax-6o 2137

This theorem is referenced by:  a5i-o  2150  nfa1-o  2166  ax67to6  2167  ax467to6  2171  dvelimf-o  2180  ax11indalem  2197  ax11inda2ALT  2198  ax11inda  2200