Theorem dveeq1 2007

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 1803	. 2 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
2		equcom 1694	. 2 |- ( z = y <-> y = z )
3	2	albii 1458	. 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 1341

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 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbal2  2008