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Theorem csboprabg 33306
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 4041 . . 3 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
2 sbcex2 3519 . . . . 5 ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
3 sbcex2 3519 . . . . . . 7 ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
4 sbcex2 3519 . . . . . . . . 9 ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
5 sbcan 3511 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑))
6 sbcg 3536 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ↔ 𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩))
76anbi1d 741 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
85, 7syl5bb 272 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
98exbidv 1890 . . . . . . . . 9 (𝐴𝑉 → (∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
104, 9syl5bb 272 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1110exbidv 1890 . . . . . . 7 (𝐴𝑉 → (∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
123, 11syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1312exbidv 1890 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
142, 13syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1514abbidv 2770 . . 3 (𝐴𝑉 → {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
161, 15syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
17 df-oprab 6694 . . 3 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
1817csbeq2i 4026 . 2 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
19 df-oprab 6694 . 2 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)}
2016, 18, 193eqtr4g 2710 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 1523  wex 1744  wcel 2030  {cab 2637  [wsbc 3468  csb 3566  cop 4216  {coprab 6691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949  df-oprab 6694
This theorem is referenced by:  csbmpt22g  33307
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