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Theorem cbv3hv 1878
Description: Lemma for ax10 2025. Similar to cbv3h 1972. Requires distinct variables but avoids ax-12 1950. (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 1568 . 2  |-  ( A. x ph  ->  A. x A. y ph )
3 a9ev 1668 . . . . . . 7  |-  E. x  x  =  y
4 cbv3hv.3 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4eximii 1587 . . . . . 6  |-  E. x
( ph  ->  ps )
6519.35i 1611 . . . . 5  |-  ( A. x ph  ->  E. x ps )
7 cbv3hv.2 . . . . . 6  |-  ( ps 
->  A. x ps )
8719.9h 1794 . . . . 5  |-  ( E. x ps  <->  ps )
96, 8sylib 189 . . . 4  |-  ( A. x ph  ->  ps )
109alimi 1568 . . 3  |-  ( A. y A. x ph  ->  A. y ps )
1110a7s 1750 . 2  |-  ( A. x A. y ph  ->  A. y ps )
122, 11syl 16 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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