Theorem dveeq2 1880

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)

Assertion

Ref	Expression
dveeq2	|- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem dveeq2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1649	. 2 |- ( w = y -> ( z = w <-> z = y ) )
2	1	dvelimv 1879	1 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1527

This theorem is referenced by:  ax10  1884  ax9  1889  ax11v2  1932  sbal1  2065  copsexg  4252  axpowndlem3  8221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529