Theorem dveel1 1893

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 1416	. 2 |- (w e. z -> A.x w e. z)
2		ax-17 1416	. 2 |- (y e. z -> A.w y e. z)
3		elequ1 1597	. 2 |- (w = y -> (w e. z <-> y e. z))
4	1, 2, 3	dvelimf 1888	1 |- (-. A.x x = y -> (y e. z -> A.x y e. z))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by: (None)