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Theorem csbcnv 5184
Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5185 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 4500 . . . 4  |-  ( [. A  /  x ]. z F y  <->  z [_ A  /  x ]_ F
y )
21opabbii 4511 . . 3  |-  { <. y ,  z >.  |  [. A  /  x ]. z F y }  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
3 csbopab 4779 . . 3  |-  [_ A  /  x ]_ { <. y ,  z >.  |  z F y }  =  { <. y ,  z
>.  |  [. A  /  x ]. z F y }
4 df-cnv 5007 . . 3  |-  `' [_ A  /  x ]_ F  =  { <. y ,  z
>.  |  z [_ A  /  x ]_ F
y }
52, 3, 43eqtr4ri 2507 . 2  |-  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
6 df-cnv 5007 . . 3  |-  `' F  =  { <. y ,  z
>.  |  z F
y }
76csbeq2i 3836 . 2  |-  [_ A  /  x ]_ `' F  =  [_ A  /  x ]_ { <. y ,  z
>.  |  z F
y }
85, 7eqtr4i 2499 1  |-  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   [.wsbc 3331   [_csb 3435   class class class wbr 4447   {copab 4504   `'ccnv 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007
This theorem is referenced by: (None)
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