Theorem wl-ax11-lem1 33362

Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem1	|- ( A. x x = y -> ( A. x x = z <-> A. y y = z ) )

Proof of Theorem wl-ax11-lem1

Step	Hyp	Ref	Expression
1		wl-aetr 33317	. 2 |- ( A. x x = y -> ( A. x x = z -> A. y y = z ) )
2		wl-aetr 33317	. . 3 |- ( A. y y = x -> ( A. y y = z -> A. x x = z ) )
3	2	aecoms 2312	. 2 |- ( A. x x = y -> ( A. y y = z -> A. x x = z ) )
4	1, 3	impbid 202	1 |- ( A. x x = y -> ( A. x x = z <-> A. y y = z ) )

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

This theorem is referenced by:  wl-ax11-lem8  33369