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Theorem sb4e 1805
Description: One direction of a simplified definition of substitution that unlike sb4 1832 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4e  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem sb4e
StepHypRef Expression
1 sb1 1766 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5e 1795 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
31, 2syl 14 1  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351   E.wex 1492   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-11 1506  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  hbsb2e  1807
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