Theorem dveeq1 2048

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 1839	. 2 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
2		equcom 1730	. 2 |- ( z = y <-> y = z )
3	2	albii 1494	. 2 |- ( A. x z = y <-> A. x y = z )
4	1, 2, 3	3imtr3g 204	1 |- ( -. A. x x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbal2  2049