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Theorem csbcnv 5741
Description: Move class substitution in and out of the converse of a relation. Version of csbcnvgALT 5742 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 / 𝑥𝐹 = 𝐴 / 𝑥𝐹

Proof of Theorem csbcnv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr 5107 . . . 4 ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦)
21opabbii 5119 . . 3 {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
3 csbopab 5429 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}
4 df-cnv 5550 . . 3 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
52, 3, 43eqtr4ri 2858 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
6 df-cnv 5550 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
76csbeq2i 3874 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
85, 7eqtr4i 2850 1 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  [wsbc 3758  csb 3866   class class class wbr 5052  {copab 5114  ccnv 5541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-cnv 5550
This theorem is referenced by:  esum2dlem  31412  csbpredg  34692
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