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Theorem csbcnvgALT 5217
Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5216 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcnvgALT (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Proof of Theorem csbcnvgALT
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr123 4631 . . . . 5 ([𝐴 / 𝑥]𝑧𝐹𝑦𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
2 csbconstg 3512 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
3 csbconstg 3512 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
42, 3breq12d 4591 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑧𝐴 / 𝑥𝐹𝑦))
51, 4syl5bb 271 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦))
65opabbidv 4643 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
7 csbopabgALT 4923 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8 df-cnv 5036 . . . 4 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
98a1i 11 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
106, 7, 93eqtr4rd 2655 . 2 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11 df-cnv 5036 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1211csbeq2i 3945 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1310, 12syl6eqr 2662 1 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  [wsbc 3402  csb 3499   class class class wbr 4578  {copab 4637  ccnv 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-cnv 5036
This theorem is referenced by: (None)
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