Theorem ax11o 1653

Description: Derivation of set.mm's original ax-11o 1654 from the shorter ax-11 1389 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1644 or ax-17 1402.

Another open problem is whether this theorem can be proved without relying on ax-12 1393 (see note in a12study 1825).

Theorem ax11 1655 shows the reverse derivation of ax-11 1389 from ax-11o 1654.

Normally, ax11o 1653 should be used rather than ax-11o 1654, except by theorems specifically studying the latter's properties.

Assertion

Ref	Expression
ax11o	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Proof of Theorem ax11o

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-11 1389	. 2 |- (x = z -> (A.zph -> A.x(x = z -> ph)))
2	1	ax11a2 1652	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1335

This theorem is referenced by:  equs5  1657  ax11v  1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ia1 98  ax-ia2 99  ax-ia3 100  ax-in1 527  ax-in2 528  ax-io 607  ax-5 1336  ax-6 1337  ax-7 1338  ax-gen 1339  ax-ie1 1375  ax-ie2 1376  ax-8 1387  ax-10 1388  ax-11 1389  ax-i12 1391  ax-4 1392  ax-17 1402  ax-i9 1417  ax-ial 1430  ax-i5r 1431

This theorem depends on definitions:  df-bi 109