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Theorem frege60b 36572
Description: Swap antecedents of ax-frege58b 36568. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege60b  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( [
y  /  x ] ps  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) )

Proof of Theorem frege60b
StepHypRef Expression
1 ax-frege58b 36568 . . 3  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  [ y  /  x ] ( ph  ->  ( ps  ->  ch ) ) )
2 sbim 2244 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ( ps 
->  ch ) )  <->  ( [
y  /  x ] ph  ->  [ y  /  x ] ( ps  ->  ch ) ) )
3 sbim 2244 . . . . 5  |-  ( [ y  /  x ]
( ps  ->  ch ) 
<->  ( [ y  /  x ] ps  ->  [ y  /  x ] ch ) )
43imbi2i 319 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ( ps 
->  ch ) )  <->  ( [
y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] ch ) ) )
52, 4bitri 257 . . 3  |-  ( [ y  /  x ]
( ph  ->  ( ps 
->  ch ) )  <->  ( [
y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] ch ) ) )
61, 5sylib 201 . 2  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( [
y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] ch ) ) )
7 frege12 36480 . 2  |-  ( ( A. x ( ph  ->  ( ps  ->  ch ) )  ->  ( [ y  /  x ] ph  ->  ( [
y  /  x ] ps  ->  [ y  /  x ] ch ) ) )  ->  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( [
y  /  x ] ps  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) ) )
86, 7ax-mp 5 1  |-  ( A. x ( ph  ->  ( ps  ->  ch )
)  ->  ( [
y  /  x ] ps  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450   [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  ax-frege1 36457  ax-frege2 36458  ax-frege8 36476  ax-frege58b 36568
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: (None)
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