Theorem dveel1 2370

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		elequ1 1997	. 2 |- ( w = y -> ( w e. z <-> y e. z ) )
2	1	dvelimv 2338	1 |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  distel  31709