Theorem dveeq2 2146

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 1930. (Revised by Wolf Lammen, 8-Sep-2018.)

Assertion

Ref	Expression
dveeq2	|- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem dveeq2

Step	Hyp	Ref	Expression
1		nfeqf2 2145	. 2 |- ( -. A. x x = y -> F/ x z = y )
2	1	nfrd 1963	1 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678

This theorem is referenced by:  ax12v2  2185  wl-ax12v3  31956