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Theorem csboprabg 32352
 Description: Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csboprabg (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
Distinct variable groups:   𝐴,𝑑   𝑦,𝐴   𝑧,𝐴   𝑉,𝑑   𝑦,𝑉   𝑧,𝑉   𝑥,𝑑   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑑)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem csboprabg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 csbab 3960 . . 3 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
2 sbcex2 3453 . . . . 5 ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
3 sbcex2 3453 . . . . . . 7 ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
4 sbcex2 3453 . . . . . . . . 9 ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
5 sbcan 3445 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑))
6 sbcg 3470 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ↔ 𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩))
76anbi1d 737 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
85, 7syl5bb 271 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
98exbidv 1837 . . . . . . . . 9 (𝐴𝑉 → (∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
104, 9syl5bb 271 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1110exbidv 1837 . . . . . . 7 (𝐴𝑉 → (∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
123, 11syl5bb 271 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1312exbidv 1837 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
142, 13syl5bb 271 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1514abbidv 2728 . . 3 (𝐴𝑉 → {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
161, 15syl5eq 2656 . 2 (𝐴𝑉𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
17 df-oprab 6553 . . 3 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
1817csbeq2i 3945 . 2 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
19 df-oprab 6553 . 2 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)}
2016, 18, 193eqtr4g 2669 1 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499  ⟨cop 4131  {coprab 6550 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-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-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875  df-oprab 6553 This theorem is referenced by:  csbmpt22g  32353
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