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Theorem dvelimv 2075
Description: Similar to dvelim 2074 with first hypothesis replaced by 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
StepHypRef Expression
1 ax-17 1627 . 2  |-  ( ph  ->  A. x ph )
2 dvelimv.1 . 2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
31, 2dvelim 2074 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  dveeq1OLD  2077  dveeq2  2078  dveel1  2107  dveel2  2108  rgen2a  2774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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