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Theorem sblim 2246
Description: Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sblim.1  |-  F/ x ps
Assertion
Ref Expression
sblim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)

Proof of Theorem sblim
StepHypRef Expression
1 sbim 2244 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sblim.1 . . . 4  |-  F/ x ps
32sbf 2229 . . 3  |-  ( [ y  /  x ] ps 
<->  ps )
43imbi2i 319 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
51, 4bitri 257 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   F/wnf 1675   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbmo  2364
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