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Theorem sbimi 1663
Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
Hypothesis
Ref Expression
sbimi.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
sbimi  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )

Proof of Theorem sbimi
StepHypRef Expression
1 sbimi.1 . . . 4  |-  ( ph  ->  ps )
21imim2i 12 . . 3  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) )
31anim2i 328 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  ps )
)
43eximi 1507 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ps ) )
52, 4anim12i 325 . 2  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ( (
x  =  y  ->  ps )  /\  E. x
( x  =  y  /\  ps ) ) )
6 df-sb 1662 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
7 df-sb 1662 . 2  |-  ( [ y  /  x ] ps 
<->  ( ( x  =  y  ->  ps )  /\  E. x ( x  =  y  /\  ps ) ) )
85, 6, 73imtr4i 194 1  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   E.wex 1397   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbbii  1664  sb6f  1700  hbsb3  1705  sbidm  1747  sbco  1858  sbcocom  1860  elsb3  1868  elsb4  1869  sbalyz  1891  hbsb4t  1905  moimv  1982  oprcl  3601  peano1  4345  peano2  4346
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