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Theorem hbsb2e 2153
Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
hbsb2e  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )

Proof of Theorem hbsb2e
StepHypRef Expression
1 sb4e 2051 . 2  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )
2 sb2 2144 . . 3  |-  ( A. x ( x  =  y  ->  E. y ph )  ->  [ y  /  x ] E. y ph )
32axc4i 1952 . 2  |-  ( A. x ( x  =  y  ->  E. y ph )  ->  A. x [ y  /  x ] E. y ph )
41, 3syl 17 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   E.wex 1659   [wsb 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by: (None)
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