Theorem hbae-o 32181

Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2111 using ax-c11 32167. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbae-o	|- ( A. x x = y -> A. z A. x x = y )

Proof of Theorem hbae-o

Step	Hyp	Ref	Expression
1		ax-c5 32163	. . . . 5 |- ( A. x x = y -> x = y )
2		ax-c9 32170	. . . . 5 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
3	1, 2	syl7 70	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( A. x x = y -> A. z x = y ) ) )
4		ax-c11 32167	. . . . 5 |- ( A. x x = z -> ( A. x x = y -> A. z x = y ) )
5	4	aecoms-o 32180	. . . 4 |- ( A. z z = x -> ( A. x x = y -> A. z x = y ) )
6		ax-c11 32167	. . . . . . 7 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
7	6	pm2.43i 49	. . . . . 6 |- ( A. x x = y -> A. y x = y )
8		ax-c11 32167	. . . . . 6 |- ( A. y y = z -> ( A. y x = y -> A. z x = y ) )
9	7, 8	syl5 33	. . . . 5 |- ( A. y y = z -> ( A. x x = y -> A. z x = y ) )
10	9	aecoms-o 32180	. . . 4 |- ( A. z z = y -> ( A. x x = y -> A. z x = y ) )
11	3, 5, 10	pm2.61ii 168	. . 3 |- ( A. x x = y -> A. z x = y )
12	11	axc4i-o 32178	. 2 |- ( A. x x = y -> A. x A. z x = y )
13		ax-11 1894	. 2 |- ( A. x A. z x = y -> A. z A. x x = y )
14	12, 13	syl 17	1 |- ( A. x x = y -> A. z A. x x = y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-11 1894  ax-c5 32163  ax-c4 32164  ax-c7 32165  ax-c11 32167  ax-c9 32170

This theorem depends on definitions:  df-bi 188  df-ex 1660

This theorem is referenced by:  dral1-o  32182  hbnae-o  32207  dral2-o  32209  aev-o  32210