Theorem equvelv 1963

Description: A specialized version of equvel 2347 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.)

Assertion

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

Distinct variable groups:   x, z   y, z

Proof of Theorem equvelv

Step	Hyp	Ref	Expression
1		equtrr 1949	. . 3 |- ( x = y -> ( z = x -> z = y ) )
2	1	alrimiv 1855	. 2 |- ( x = y -> A. z ( z = x -> z = y ) )
3		equs4v 1930	. . 3 |- ( A. z ( z = x -> z = y ) -> E. z ( z = x /\ z = y ) )
4		equvinv 1959	. . 3 |- ( x = y <-> E. z ( z = x /\ z = y ) )
5	3, 4	sylibr 224	. 2 |- ( A. z ( z = x -> z = y ) -> x = y )
6	2, 5	impbii 199	1 |- ( x = y <-> A. z ( z = x -> z = y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)