Theorem ax11 2155

Description: Rederivation of Axiom ax-11 1746 from ax-11o 2141, ax-10o 2139, and other older axioms. The proof does not require ax-16 2144 or ax-17 1616. See Theorem ax11o 1994 for the derivation of ax-11o 2141 from ax-11 1746.

An open problem is whether we can prove this using ax-10 2140 instead of ax-10o 2139.

This proof uses newer axioms ax-5 1557 and ax-9 1654, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2136 and ax-9o 2138. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax11	⊢ (x = y → (∀yφ → ∀x(x = y → φ)))

Proof of Theorem ax11

Step	Hyp	Ref	Expression
1		biidd 228	. . . . 5 ⊢ (∀x x = y → (φ ↔ φ))
2	1	dral1-o 2154	. . . 4 ⊢ (∀x x = y → (∀xφ ↔ ∀yφ))
3		ax-1 6	. . . . 5 ⊢ (φ → (x = y → φ))
4	3	alimi 1559	. . . 4 ⊢ (∀xφ → ∀x(x = y → φ))
5	2, 4	syl6bir 220	. . 3 ⊢ (∀x x = y → (∀yφ → ∀x(x = y → φ)))
6	5	a1d 22	. 2 ⊢ (∀x x = y → (x = y → (∀yφ → ∀x(x = y → φ))))
7		ax-4 2135	. . 3 ⊢ (∀yφ → φ)
8		ax-11o 2141	. . 3 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
9	7, 8	syl7 63	. 2 ⊢ (¬ ∀x x = y → (x = y → (∀yφ → ∀x(x = y → φ))))
10	6, 9	pm2.61i 156	1 ⊢ (x = y → (∀yφ → ∀x(x = y → 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-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-11o 2141  ax-12o 2142

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

This theorem is referenced by:  ax10o-o  2203