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Theorem csbcnv 5211
Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5212 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 4626 . . . 4 ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦)
21opabbii 4638 . . 3 {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
3 csbopab 4917 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}
4 df-cnv 5031 . . 3 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
52, 3, 43eqtr4ri 2637 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
6 df-cnv 5031 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
76csbeq2i 3939 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
85, 7eqtr4i 2629 1 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  [wsbc 3396  csb 3493   class class class wbr 4572  {copab 4631  ccnv 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-br 4573  df-opab 4633  df-cnv 5031
This theorem is referenced by:  esum2dlem  29282  csbpredg  32146
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