Theorem ax13v 2247

Description: A weaker version of ax-13 2246 with distinct variable restrictions on pairs x , z and y , z. In order to show (with ax13 2249) that this weakening is still adequate, this should be the only theorem referencing ax-13 2246 directly.

Had we additionally required x and y be distinct, too, this theorem would have been a direct consequence of ax-5 1839. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. (Contributed by NM, 30-Jun-2016.)

Assertion

Ref	Expression
ax13v	|- ( -. x = y -> ( y = z -> A. x y = z ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem ax13v

Step	Hyp	Ref	Expression
1		ax-13 2246	1 |- ( -. x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-13 2246

This theorem is referenced by:  ax13lem1  2248  wl-spae  33306