Theorem sbeqalbi 38601

Description: When both x and z and y and z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)

Assertion

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

Distinct variable groups:   y, z   x, z

Proof of Theorem sbeqalbi

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		sbeqal1 38598	. 2 |- ( A. z ( z = x -> z = y ) -> x = y )
4	2, 3	impbii 199	1 |- ( x = y <-> A. z ( z = x -> z = y ) )

Syntax hints:   -> 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-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)