Theorem ax10o 1181

Description: Show that ax-10o 1182 can be derived from ax-10 1007. An open problem is whether this theorem can be derived from ax-10 1007 and the others when ax-11 1008 is replaced with ax-11o 1260. See theorem ax10 1183 for the rederivation of ax-10 1007 from ax10o 1181.

This theorem should not be referenced in any proof. Instead, use ax-10o 1182 below so that uses of ax-10o 1182 can be more easily identified.

Assertion

Ref	Expression
ax10o	|- (A.x x = y -> (A.xph -> A.yph))

Proof of Theorem ax10o

Step	Hyp	Ref	Expression
1		ax-11 1008	. . . 4 |- (y = x -> (A.xph -> A.y(y = x -> ph)))
2	1	equcoms 1172	. . 3 |- (x = y -> (A.xph -> A.y(y = x -> ph)))
3	2	a4s 1025	. 2 |- (A.x x = y -> (A.xph -> A.y(y = x -> ph)))
4		ax-10 1007	. . 3 |- (A.x x = y -> A.y y = x)
5		pm2.27 62	. . . 4 |- (y = x -> ((y = x -> ph) -> ph))
6	5	19.20ii 1036	. . 3 |- (A.y y = x -> (A.y(y = x -> ph) -> A.yph))
7	4, 6	syl 10	. 2 |- (A.x x = y -> (A.y(y = x -> ph) -> A.yph))
8	3, 7	syld 27	1 |- (A.x x = y -> (A.xph -> A.yph))

Syntax hints:   -> wi 3  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164

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