Theorem hbae 2056

Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Assertion

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

Proof of Theorem hbae

Step	Hyp	Ref	Expression
1		sp 1860	. . . . 5 |- ( A. x x = y -> x = y )
2		axc9 2047	. . . . 5 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
3	1, 2	syl7 68	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( A. x x = y -> A. z x = y ) ) )
4		axc112 1938	. . . 4 |- ( A. z z = x -> ( A. x x = y -> A. z x = y ) )
5		axc11 2055	. . . . . 6 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
6	5	pm2.43i 47	. . . . 5 |- ( A. x x = y -> A. y x = y )
7		axc112 1938	. . . . 5 |- ( A. z z = y -> ( A. y x = y -> A. z x = y ) )
8	6, 7	syl5 32	. . . 4 |- ( A. z z = y -> ( A. x x = y -> A. z x = y ) )
9	3, 4, 8	pm2.61ii 165	. . 3 |- ( A. x x = y -> A. z x = y )
10	9	axc4i 1899	. 2 |- ( A. x x = y -> A. x A. z x = y )
11		ax-11 1843	. 2 |- ( A. x A. z x = y -> A. z A. x x = y )
12	10, 11	syl 16	1 |- ( A. x x = y -> A. z A. x x = y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618

This theorem is referenced by:  nfae  2057  hbnae  2058  aevALT  2064  drex2  2071  ax6e2eq  33473