Theorem ax9o 1950

Description: Show that the original axiom ax-9o 2138 can be derived from ax9 1949 and others. See ax9from9o 2148 for the rederivation of ax9 1949 from ax-9o 2138.

Normally, ax9o 1950 should be used rather than ax-9o 2138, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax9o	⊢ (∀x(x = y → ∀xφ) → φ)

Proof of Theorem ax9o

Step	Hyp	Ref	Expression
1		ax9 1949	. . 3 ⊢ ¬ ∀x ¬ x = y
2		con3 126	. . . 4 ⊢ ((x = y → ∀xφ) → (¬ ∀xφ → ¬ x = y))
3	2	al2imi 1561	. . 3 ⊢ (∀x(x = y → ∀xφ) → (∀x ¬ ∀xφ → ∀x ¬ x = y))
4	1, 3	mtoi 169	. 2 ⊢ (∀x(x = y → ∀xφ) → ¬ ∀x ¬ ∀xφ)
5		ax6o 1750	. 2 ⊢ (¬ ∀x ¬ ∀xφ → φ)
6	4, 5	syl 15	1 ⊢ (∀x(x = y → ∀xφ) → φ)

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-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  equsal  1960  spimt  1974  cbv1h  1978