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Theorem sbequ1 2084
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.)
Assertion
Ref Expression
sbequ1  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )

Proof of Theorem sbequ1
StepHypRef Expression
1 pm3.4 564 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  ->  ph )
)
2 19.8a 1937 . . 3  |-  ( ( x  =  y  /\  ph )  ->  E. x
( x  =  y  /\  ph ) )
3 df-sb 1800 . . 3  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
41, 2, 3sylanbrc 671 . 2  |-  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
54ex 436 1  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   E.wex 1665   [wsb 1799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-12 1935
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-sb 1800
This theorem is referenced by:  sbequ12  2085  dfsb2  2204  sbequi  2206  sbi1  2223  2eu6  2389  sb5ALT  36893  2pm13.193  36930  2pm13.193VD  37310  sb5ALTVD  37320
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