Theorem dveel2 1938

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

Assertion

Ref	Expression
dveel2	|- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )

Distinct variable group:   x, z

Proof of Theorem dveel2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1459	. 2 |- ( z e. w -> A. x z e. w )
2		ax-17 1459	. 2 |- ( z e. y -> A. w z e. y )
3		elequ2 1641	. 2 |- ( w = y -> ( z e. w <-> z e. y ) )
4	1, 2, 3	dvelimf 1932	1 |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by: (None)