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Theorem cbv3hv 2080
Description: Version of cbv3h 2122 with a dv condition on  x ,  y, which does not require ax-13 2104. Was used in a proof of axc11n 2157 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
Hypotheses
Ref Expression
cbv3hv.nf1  |-  ( ph  ->  A. y ph )
cbv3hv.nf2  |-  ( ps 
->  A. x ps )
cbv3hv.1  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3hv  |-  ( A. x ph  ->  A. y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbv3hv
StepHypRef Expression
1 cbv3hv.nf1 . . 3  |-  ( ph  ->  A. y ph )
21nfi 1682 . 2  |-  F/ y
ph
3 cbv3hv.nf2 . . 3  |-  ( ps 
->  A. x ps )
43nfi 1682 . 2  |-  F/ x ps
5 cbv3hv.1 . 2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
62, 4, 5cbv3v 2078 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
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