Theorem ax5o 1749

Description: Show that the original axiom ax-5o 2136 can be derived from ax-5 1557 and others. See ax5 2146 for the rederivation of ax-5 1557 from ax-5o 2136.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax5o	⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))

Proof of Theorem ax5o

Step	Hyp	Ref	Expression
1		sp 1747	. . . 4 ⊢ (∀x ¬ ∀xφ → ¬ ∀xφ)
2	1	con2i 112	. . 3 ⊢ (∀xφ → ¬ ∀x ¬ ∀xφ)
3		hbn1 1730	. . 3 ⊢ (¬ ∀x ¬ ∀xφ → ∀x ¬ ∀x ¬ ∀xφ)
4		hbn1 1730	. . . . 5 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
5	4	con1i 121	. . . 4 ⊢ (¬ ∀x ¬ ∀xφ → ∀xφ)
6	5	alimi 1559	. . 3 ⊢ (∀x ¬ ∀x ¬ ∀xφ → ∀x∀xφ)
7	2, 3, 6	3syl 18	. 2 ⊢ (∀xφ → ∀x∀xφ)
8		ax-5 1557	. 2 ⊢ (∀x(∀xφ → ψ) → (∀x∀xφ → ∀xψ))
9	7, 8	syl5 28	1 ⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1540

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

This theorem is used by: (None)