Theorem ax10o 1767

Description: Show that ax-10o 1768 can be derived from ax-10 1558. An open problem is whether this theorem can be derived from ax-10 1558 and the others when ax-11 1559 is replaced with ax-11o 1876. See Theorem ax10 1769 for the rederivation of ax-10 1558 from ax10o 1767.

Normally, ax10o 1767 should be used rather than ax-10o 1768, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Assertion

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

Proof of Theorem ax10o

Step	Hyp	Ref	Expression
1		ax-10 1558	. 2 |- ( A. x x = y -> A. y y = x )
2		ax-11 1559	. . . 4 |- ( y = x -> ( A. x ph -> A. y ( y = x -> ph ) ) )
3	2	equcoms 1760	. . 3 |- ( x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) )
4	3	sps 1590	. 2 |- ( A. x x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) )
5		pm2.27 40	. . 3 |- ( y = x -> ( ( y = x -> ph ) -> ph ) )
6	5	al2imi 1511	. 2 |- ( A. y y = x -> ( A. y ( y = x -> ph ) -> A. y ph ) )
7	1, 4, 6	sylsyld 58	1 |- ( A. x x = y -> ( A. x ph -> A. y ph ) )

Syntax hints:   -> wi 4   A.wal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbae  1770  dral1  1783