Theorem dvelimv 2338

Description: Similar to dvelim 2337 with first hypothesis replaced by a distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

Hypothesis

Ref	Expression
dvelimv.1	|- ( z = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
dvelimv	|- ( -. A. x x = y -> ( ps -> A. x ps ) )

Distinct variable groups:   ph, x   ps, z

Allowed substitution hints:   ph( y, z)   ps( x, y)

Proof of Theorem dvelimv

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 |- ( ph -> A. x ph )
2		dvelimv.1	. 2 |- ( z = y -> ( ph <-> ps ) )
3	1, 2	dvelim 2337	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  dveeq2ALT  2340  dveel1  2370  dveel2  2371  rgen2a  2977