Theorem ax11o 1994

Description: Derivation of set.mm's original ax-11o 2141 from ax-10 2140 and the shorter ax-11 1746 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2144 or ax-17 1616 (given all of the original and new versions of sp 1747 through ax-15 2143).

Another open problem is whether this theorem can be proved without relying on ax12o 1934.

Theorem ax11 2155 shows the reverse derivation of ax-11 1746 from ax-11o 2141.

Normally, ax11o 1994 should be used rather than ax-11o 2141, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Assertion

Ref	Expression
ax11o	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Proof of Theorem ax11o

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-11 1746	. 2 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
2	1	ax11a2 1993	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀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-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:  ax11b  1995  equs5  1996  ax11v  2096