Theorem ax9o 1163

Description: Show that the original axiom ax-9o 1164 can be derived from ax-9 1006 and others. See ax9 1165 for the rederivation of ax-9 1006 from ax-9o 1164.

This theorem should not be referenced in any proof. Instead, use ax-9o 1164 below so that uses of ax-9o 1164 can be more easily identified.

Assertion

Ref	Expression
ax9o	|- (A.x(x = y -> A.xph) -> ph)

Proof of Theorem ax9o

Step	Hyp	Ref	Expression
1		ax-9 1006	. . . 4 |- -. A.x -. x = y
2		df-ex 1022	. . . 4 |- (E.x x = y <-> -. A.x -. x = y)
3	1, 2	mpbir 197	. . 3 |- E.x x = y
4		19.22 1080	. . 3 |- (A.x(x = y -> A.xph) -> (E.x x = y -> E.xA.xph))
5	3, 4	mpi 44	. 2 |- (A.x(x = y -> A.xph) -> E.xA.xph)
6		a6e 1031	. 2 |- (E.xA.xph -> ph)
7	5, 6	syl 10	1 |- (A.x(x = y -> A.xph) -> ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 995   = wceq 997  E.wex 1021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-9 1006  ax-4 1014  ax-5o 1016  ax-6o 1019

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022

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