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Theorem opelopabsb 4061
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem opelopabsb
Dummy variables 𝑣 𝑢 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elopab 4059 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} ↔ ∃𝑢𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2 simpl 107 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩)
32eqcomd 2090 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
4 vex 2618 . . . . . . . 8 𝑢 ∈ V
5 vex 2618 . . . . . . . 8 𝑣 ∈ V
64, 5opth 4038 . . . . . . 7 (⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑢 = 𝐴𝑣 = 𝐵))
73, 6sylib 120 . . . . . 6 ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑢 = 𝐴𝑣 = 𝐵))
872eximi 1535 . . . . 5 (∃𝑢𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ∃𝑢𝑣(𝑢 = 𝐴𝑣 = 𝐵))
9 eeanv 1852 . . . . . 6 (∃𝑢𝑣(𝑢 = 𝐴𝑣 = 𝐵) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
10 isset 2619 . . . . . . 7 (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
11 isset 2619 . . . . . . 7 (𝐵 ∈ V ↔ ∃𝑣 𝑣 = 𝐵)
1210, 11anbi12i 448 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
139, 12bitr4i 185 . . . . 5 (∃𝑢𝑣(𝑢 = 𝐴𝑣 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
148, 13sylib 120 . . . 4 (∃𝑢𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
151, 14sylbi 119 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16 nfv 1464 . . . 4 𝑢𝜑
17 nfv 1464 . . . 4 𝑣𝜑
18 nfs1v 1860 . . . 4 𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
19 nfs1v 1860 . . . . 5 𝑦[𝑣 / 𝑦]𝜑
2019nfsbxy 1863 . . . 4 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
21 sbequ12 1698 . . . . 5 (𝑦 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑦]𝜑))
22 sbequ12 1698 . . . . 5 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2321, 22sylan9bbr 451 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2416, 17, 18, 20, 23cbvopab 3884 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑}
2515, 24eleq2s 2179 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
26 sbcex 2837 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝐴 ∈ V)
27 spesbc 2913 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
28 sbcex 2837 . . . . 5 ([𝐵 / 𝑦]𝜑𝐵 ∈ V)
2928exlimiv 1532 . . . 4 (∃𝑥[𝐵 / 𝑦]𝜑𝐵 ∈ V)
3027, 29syl 14 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝐵 ∈ V)
3126, 30jca 300 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32 opeq1 3605 . . . . 5 (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
3332eleq1d 2153 . . . 4 (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
34 dfsbcq2 2832 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
3533, 34bibi12d 233 . . 3 (𝑧 = 𝐴 → ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)))
36 opeq2 3606 . . . . 5 (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
3736eleq1d 2153 . . . 4 (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
38 dfsbcq2 2832 . . . . 5 (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑))
3938sbcbidv 2886 . . . 4 (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
4037, 39bibi12d 233 . . 3 (𝑤 = 𝐵 → ((⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)))
41 nfopab1 3882 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4241nfel2 2237 . . . . 5 𝑥𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43 nfs1v 1860 . . . . 5 𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
4442, 43nfbi 1524 . . . 4 𝑥(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
45 opeq1 3605 . . . . . 6 (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
4645eleq1d 2153 . . . . 5 (𝑥 = 𝑧 → (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
47 sbequ12 1698 . . . . 5 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
4846, 47bibi12d 233 . . . 4 (𝑥 = 𝑧 → ((⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
49 nfopab2 3883 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
5049nfel2 2237 . . . . . 6 𝑦𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
51 nfs1v 1860 . . . . . 6 𝑦[𝑤 / 𝑦]𝜑
5250, 51nfbi 1524 . . . . 5 𝑦(⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
53 opeq2 3606 . . . . . . 7 (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
5453eleq1d 2153 . . . . . 6 (𝑦 = 𝑤 → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
55 sbequ12 1698 . . . . . 6 (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
5654, 55bibi12d 233 . . . . 5 (𝑦 = 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) ↔ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)))
57 opabid 4058 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
5852, 56, 57chvar 1684 . . . 4 (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
5944, 48, 58chvar 1684 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
6035, 40, 59vtocl2g 2676 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
6125, 31, 60pm5.21nii 653 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  [wsb 1689  Vcvv 2615  [wsbc 2829  cop 3434  {copab 3873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3875
This theorem is referenced by:  brabsb  4062  opelopabaf  4074  opelopabf  4075  difopab  4537  isarep1  5065
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