Theorem dveel1 1995

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
dveel1	|- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )

Distinct variable group:   x, z

Proof of Theorem dveel1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1506	. 2 |- ( w e. z -> A. x w e. z )
2		ax-17 1506	. 2 |- ( y e. z -> A. w y e. z )
3		elequ1 1690	. 2 |- ( w = y -> ( w e. z <-> y e. z ) )
4	1, 2, 3	dvelimf 1990	1 |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1329

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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by: (None)