Theorem dveeq1 2012

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
dveeq1	|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )

Distinct variable group:   x, z

Proof of Theorem dveeq1

Step	Hyp	Ref	Expression
1		dveeq2 1808	. 2 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
2		equcom 1699	. 2 |- ( z = y <-> y = z )
3	2	albii 1463	. 2 |- ( A. x z = y <-> A. x y = z )
4	1, 2, 3	3imtr3g 203	1 |- ( -. A. x x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbal2  2013