Theorem dveel2 1897

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 1419	. 2 |- ( z e. w -> A. x z e. w )
2		ax-17 1419	. 2 |- ( z e. y -> A. w z e. y )
3		elequ2 1601	. 2 |- ( w = y -> ( z e. w <-> z e. y ) )
4	1, 2, 3	dvelimf 1891	1 |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1241

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)