Theorem bj-hbaeb2 32805

Description: Biconditional version of a form of hbae 2315 with commuted quantifiers, not requiring ax-11 2034. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbaeb2	|- ( A. x x = y <-> A. x A. z x = y )

Proof of Theorem bj-hbaeb2

Step	Hyp	Ref	Expression
1		sp 2053	. . . . 5 |- ( A. x x = y -> x = y )
2		axc9 2302	. . . . 5 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
3	1, 2	syl7 74	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( A. x x = y -> A. z x = y ) ) )
4		axc11r 2187	. . . 4 |- ( A. z z = x -> ( A. x x = y -> A. z x = y ) )
5		axc11 2314	. . . . . 6 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
6	5	pm2.43i 52	. . . . 5 |- ( A. x x = y -> A. y x = y )
7		axc11r 2187	. . . . 5 |- ( A. z z = y -> ( A. y x = y -> A. z x = y ) )
8	6, 7	syl5 34	. . . 4 |- ( A. z z = y -> ( A. x x = y -> A. z x = y ) )
9	3, 4, 8	pm2.61ii 177	. . 3 |- ( A. x x = y -> A. z x = y )
10	9	axc4i 2131	. 2 |- ( A. x x = y -> A. x A. z x = y )
11		sp 2053	. . 3 |- ( A. z x = y -> x = y )
12	11	alimi 1739	. 2 |- ( A. x A. z x = y -> A. x x = y )
13	10, 12	impbii 199	1 |- ( A. x x = y <-> A. x A. z x = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-hbaeb  32806  bj-dvv  32808