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Theorem dfoprab4f 5819
 Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
dfoprab4f.x 𝑥𝜑
dfoprab4f.y 𝑦𝜑
dfoprab4f.1 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
dfoprab4f {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦   𝑤,𝐵,𝑥,𝑦   𝜓,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)   𝐵(𝑧)

Proof of Theorem dfoprab4f
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . . 5 𝑥 𝑤 = ⟨𝑡, 𝑢
2 dfoprab4f.x . . . . . 6 𝑥𝜑
3 nfs1v 1815 . . . . . 6 𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
42, 3nfbi 1481 . . . . 5 𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
51, 4nfim 1464 . . . 4 𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6 opeq1 3549 . . . . . 6 (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
76eqeq2d 2051 . . . . 5 (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8 sbequ12 1654 . . . . . 6 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
98bibi2d 221 . . . . 5 (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
107, 9imbi12d 223 . . . 4 (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11 nfv 1421 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑢
12 dfoprab4f.y . . . . . . 7 𝑦𝜑
13 nfs1v 1815 . . . . . . 7 𝑦[𝑢 / 𝑦]𝜓
1412, 13nfbi 1481 . . . . . 6 𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
1511, 14nfim 1464 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16 opeq2 3550 . . . . . . 7 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
1716eqeq2d 2051 . . . . . 6 (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18 sbequ12 1654 . . . . . . 7 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
1918bibi2d 221 . . . . . 6 (𝑦 = 𝑢 → ((𝜑𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
2017, 19imbi12d 223 . . . . 5 (𝑦 = 𝑢 → ((𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓)) ↔ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))))
21 dfoprab4f.1 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
2215, 20, 21chvar 1640 . . . 4 (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
235, 10, 22chvar 1640 . . 3 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
2423dfoprab4 5818 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25 nfv 1421 . . 3 𝑡((𝑥𝐴𝑦𝐵) ∧ 𝜓)
26 nfv 1421 . . 3 𝑢((𝑥𝐴𝑦𝐵) ∧ 𝜓)
27 nfv 1421 . . . 4 𝑥(𝑡𝐴𝑢𝐵)
2827, 3nfan 1457 . . 3 𝑥((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29 nfv 1421 . . . 4 𝑦(𝑡𝐴𝑢𝐵)
3013nfsb 1822 . . . 4 𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
3129, 30nfan 1457 . . 3 𝑦((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32 eleq1 2100 . . . . 5 (𝑥 = 𝑡 → (𝑥𝐴𝑡𝐴))
33 eleq1 2100 . . . . 5 (𝑦 = 𝑢 → (𝑦𝐵𝑢𝐵))
3432, 33bi2anan9 538 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → ((𝑥𝐴𝑦𝐵) ↔ (𝑡𝐴𝑢𝐵)))
3518, 8sylan9bbr 436 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
3634, 35anbi12d 442 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
3725, 26, 28, 31, 36cbvoprab12 5578 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
3824, 37eqtr4i 2063 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  [wsb 1645  ⟨cop 3378  {copab 3817   × cxp 4343  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768 This theorem is referenced by: (None)
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