Theorem ax10o 1139

Description: Show that ax-10o 1140 can be derived from ax-10 966. An open problem is whether this theorem can be derived from ax-10 966 and the others when ax-11 967 is replaced with ax-11o 1218. See theorem ax10 1141 for the rederivation of ax-10 966 from ax10o 1139.

This theorem should not be referenced in any proof. Instead, use ax-10o 1140 below so that uses of ax-10o 1140 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-11 967	. . . 4 |- (y = x -> (A.xph -> A.y(y = x -> ph)))
2	1	equcoms 1130	. . 3 |- (x = y -> (A.xph -> A.y(y = x -> ph)))
3	2	a4s 984	. 2 |- (A.x x = y -> (A.xph -> A.y(y = x -> ph)))
4		ax-10 966	. . 3 |- (A.x x = y -> A.y y = x)
5		pm2.27 62	. . . 4 |- (y = x -> ((y = x -> ph) -> ph))
6	5	19.20ii 995	. . 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 954   = wceq 956

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

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