Theorem ax11o 1217

Description: Derivation of set.mm's original ax-11o 1218 from the shorter ax-11 967 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1210 or ax-17 971.

Another open problem is whether this theorem can be proved without relying on ax-12 968 (see note in a12study 1378).

Theorem ax11 1219 shows the reverse derivation of ax-11 967 from ax-11o 1218.

This theorem should not be referenced in any proof. Instead, use ax-11o 1218 below so that theorems needing ax-11o 1218 can be more easily identified.

Assertion

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

Proof of Theorem ax11o

Step	Hyp	Ref	Expression
1		ax-11 967	. 2 |- (x = z -> (A.zph -> A.x(x = z -> ph)))
2	1	ax11a2 1216	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 2   -> wi 3  A.wal 954   = wceq 956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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