Theorem dveel1 2157

Description: Quantifier introduction when one pair of variables is disjoint. (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 1526	. 2 |- ( w e. z -> A. x w e. z )
2		ax-17 1526	. 2 |- ( y e. z -> A. w y e. z )
3		elequ1 2152	. 2 |- ( w = y -> ( w e. z <-> y e. z ) )
4	1, 2, 3	dvelimf 2015	1 |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1351

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by: (None)