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Theorem csbcnv 5099
Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5100 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbcnv  |-  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F

Proof of Theorem csbcnv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr 4419 . . . 4  |-  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y )
21opabbii 4431 . . 3  |-  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
3 csbopab 4693 . . 3  |-  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y }
4 df-cnv 4921 . . 3  |-  `' [_ A  /  x ]_ F  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
52, 3, 43eqtr4ri 2422 . 2  |-  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
6 df-cnv 4921 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
76csbeq2i 3760 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
85, 7eqtr4i 2414 1  |-  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399   [.wsbc 3252   [_csb 3348   class class class wbr 4367   {copab 4424   `'ccnv 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-cnv 4921
This theorem is referenced by:  esum2dlem  28240
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