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Theorem bj-hbaeb2 31413
Description: Biconditional version of a form of hbae 2148 with commuted quantifiers, not requiring ax-11 1919. (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
StepHypRef Expression
1 sp 1936 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 axc9 2139 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
31, 2syl7 70 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y ) ) )
4 axc112 2019 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
5 axc11 2147 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
65pm2.43i 49 . . . . 5  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
7 axc112 2019 . . . . 5  |-  ( A. z  z  =  y  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
86, 7syl5 33 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
93, 4, 8pm2.61ii 169 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
109axc4i 1979 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
11 sp 1936 . . 3  |-  ( A. z  x  =  y  ->  x  =  y )
1211alimi 1683 . 2  |-  ( A. x A. z  x  =  y  ->  A. x  x  =  y )
1310, 12impbii 191 1  |-  ( A. x  x  =  y  <->  A. x A. z  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188   A.wal 1441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932  ax-13 2090
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667
This theorem is referenced by:  bj-hbaeb  31414  bj-dvv  31416
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