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Theorem csboprabg 35614
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 4384 . . 3 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
2 sbcex2 3792 . . . . 5 ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
3 sbcex2 3792 . . . . . . 7 ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
4 sbcex2 3792 . . . . . . . . 9 ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑))
5 sbcan 3779 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑))
6 sbcg 3806 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ↔ 𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩))
76anbi1d 630 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
85, 7bitrid 282 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ (𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
98exbidv 1923 . . . . . . . . 9 (𝐴𝑉 → (∃𝑑[𝐴 / 𝑥](𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
104, 9bitrid 282 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1110exbidv 1923 . . . . . . 7 (𝐴𝑉 → (∃𝑧[𝐴 / 𝑥]𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
123, 11bitrid 282 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1312exbidv 1923 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
142, 13bitrid 282 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)))
1514abbidv 2805 . . 3 (𝐴𝑉 → {𝑐[𝐴 / 𝑥]𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
161, 15eqtrid 2788 . 2 (𝐴𝑉𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)})
17 df-oprab 7341 . . 3 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
1817csbeq2i 3851 . 2 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = 𝐴 / 𝑥{𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ 𝜑)}
19 df-oprab 7341 . 2 {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑} = {𝑐 ∣ ∃𝑦𝑧𝑑(𝑐 = ⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∧ [𝐴 / 𝑥]𝜑)}
2016, 18, 193eqtr4g 2801 1 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wex 1780  wcel 2105  {cab 2713  [wsbc 3727  csb 3843  cop 4579  {coprab 7338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-nul 4270  df-oprab 7341
This theorem is referenced by:  csbmpo123  35615
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