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Theorem cbv3hv 1984
Description: Lemma for axc11n 2075. Similar to cbv3h 2043. Requires distinct variables but avoids ax-13 2026. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
Hypotheses
Ref Expression
cbv3hv.1  |-  ( ph  ->  A. y ph )
cbv3hv.2  |-  ( ps 
->  A. x ps )
cbv3hv.3  |-  ( 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.1 . . 3  |-  ( ph  ->  A. y ph )
21alimi 1654 . 2  |-  ( A. x ph  ->  A. x A. y ph )
3 ax6ev 1773 . . . . . . 7  |-  E. x  x  =  y
4 cbv3hv.3 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4eximii 1679 . . . . . 6  |-  E. x
( ph  ->  ps )
6519.35i 1710 . . . . 5  |-  ( A. x ph  ->  E. x ps )
7 cbv3hv.2 . . . . . 6  |-  ( ps 
->  A. x ps )
8719.9h 1917 . . . . 5  |-  ( E. x ps  <->  ps )
96, 8sylib 196 . . . 4  |-  ( A. x ph  ->  ps )
109alimi 1654 . . 3  |-  ( A. y A. x ph  ->  A. y ps )
1110alcoms 1867 . 2  |-  ( A. x A. y ph  ->  A. y ps )
122, 11syl 17 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1403   E.wex 1633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-ex 1634  df-nf 1638
This theorem is referenced by: (None)
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