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Theorem bj-hbsb2av 31366
Description: Version of hbsb2a 2190 with a dv condition, which does not require ax-13 2090. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbsb2av  |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-hbsb2av
StepHypRef Expression
1 sb4a 2086 . 2  |-  ( [ y  /  x ] A. y ph  ->  A. x
( x  =  y  ->  ph ) )
2 bj-sb2v 31358 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
32axc4i 1979 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x [ y  /  x ] ph )
41, 3syl 17 1  |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1441   [wsb 1796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667  df-sb 1797
This theorem is referenced by:  bj-hbsb3v  31367
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