Theorem ax10o 1600

Description: Show that ax-10o 1601 can be derived from ax-10 1393. An open problem is whether this theorem can be derived from ax-10 1393 and the others when ax-11 1394 is replaced with ax-11o 1701. See theorem ax10 1602 for the rederivation of ax-10 1393 from ax10o 1600.

Normally, ax10o 1600 should be used rather than ax-10o 1601, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Assertion

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

Proof of Theorem ax10o

Step	Hyp	Ref	Expression
1		ax-10 1393	. 2 |- (A.x x = y -> A.y y = x)
2		ax-11 1394	. . . 4 |- (y = x -> (A.xph -> A.y(y = x -> ph)))
3	2	equcoms 1591	. . 3 |- (x = y -> (A.xph -> A.y(y = x -> ph)))
4	3	sps 1427	. 2 |- (A.x x = y -> (A.xph -> A.y(y = x -> ph)))
5		pm2.27 35	. . 3 |- (y = x -> ((y = x -> ph) -> ph))
6	5	al2imi 1344	. 2 |- (A.y y = x -> (A.y(y = x -> ph) -> A.yph))
7	1, 4, 6	sylsyld 52	1 |- (A.x x = y -> (A.xph -> A.yph))

Syntax hints:   -> wi 4  A.wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  hbae  1603  dral1  1615