Theorem ax10o-o 2203

Description: Show that ax-10o 2139 can be derived from ax-10 2140. An open problem is whether this theorem can be derived from ax-10 2140 and the others when ax-11 1746 is replaced with ax-11o 2141. See Theorem ax10from10o 2177 for the rederivation of ax-10 2140 from ax10o 1952.

Normally, ax10o 1952 should be used rather than ax-10o 2139 or ax10o-o 2203, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10o-o	⊢ (∀x x = y → (∀xφ → ∀yφ))

Proof of Theorem ax10o-o

Step	Hyp	Ref	Expression
1		ax-10 2140	. 2 ⊢ (∀x x = y → ∀y y = x)
2		ax11 2155	. . . 4 ⊢ (y = x → (∀xφ → ∀y(y = x → φ)))
3	2	equcoms 1681	. . 3 ⊢ (x = y → (∀xφ → ∀y(y = x → φ)))
4	3	sps-o 2159	. 2 ⊢ (∀x x = y → (∀xφ → ∀y(y = x → φ)))
5		pm2.27 35	. . 3 ⊢ (y = x → ((y = x → φ) → φ))
6	5	al2imi 1561	. 2 ⊢ (∀y y = x → (∀y(y = x → φ) → ∀yφ))
7	1, 4, 6	sylsyld 52	1 ⊢ (∀x x = y → (∀xφ → ∀yφ))

Syntax hints:   → 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-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-10 2140  ax-11o 2141  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by: (None)