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Theorem sbcop 5509
Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023.)
Hypothesis
Ref Expression
sbcop.z (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
sbcop ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑎, 𝑏⟩ / 𝑧]𝜑)
Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑥,𝑦   𝜓,𝑧   𝑥,𝑏,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑎,𝑏)   𝜓(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sbcop
StepHypRef Expression
1 sbcop.z . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
21sbcop1 5508 . . 3 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
32sbcbii 3865 . 2 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑)
4 sbcnestgw 4446 . . 3 (𝑏 ∈ V → ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑))
54elv 3493 . 2 ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑)
6 csbopg 4915 . . . . 5 (𝑏 ∈ V → 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩)
76elv 3493 . . . 4 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦
8 vex 3492 . . . . . 6 𝑏 ∈ V
98csbconstgi 3943 . . . . 5 𝑏 / 𝑦𝑎 = 𝑎
108csbvargi 4458 . . . . 5 𝑏 / 𝑦𝑦 = 𝑏
119, 10opeq12i 4902 . . . 4 𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩ = ⟨𝑎, 𝑏
127, 11eqtri 2768 . . 3 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑎, 𝑏
13 dfsbcq 3806 . . 3 (𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → ([𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑[𝑎, 𝑏⟩ / 𝑧]𝜑))
1412, 13ax-mp 5 . 2 ([𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑[𝑎, 𝑏⟩ / 𝑧]𝜑)
153, 5, 143bitri 297 1 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑎, 𝑏⟩ / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  Vcvv 3488  [wsbc 3804  csb 3921  cop 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655
This theorem is referenced by:  reuop  6324
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