Theorem dveel2 2186

Description: Quantifier introduction when one pair of variables is disjoint. (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 1549	. 2 |- ( z e. w -> A. x z e. w )
2		ax-17 1549	. 2 |- ( z e. y -> A. w z e. y )
3		elequ2 2181	. 2 |- ( w = y -> ( z e. w <-> z e. y ) )
4	1, 2, 3	dvelimf 2043	1 |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1371

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 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by: (None)