Theorem dveeq2 2080

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 1700	. 2 |- ( w = y -> ( z = w <-> z = y ) )
2	1	dvelimv 2077	1 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1550

This theorem is referenced by:  ax11v2  2081  ax11v2OLD  2082  sbal1  2209  copsexg  4471  axpowndlem3  8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953

This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555