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Theorem sbcop 5494
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 5493 . . 3 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
32sbcbii 3846 . 2 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑)
4 sbcnestgw 4423 . . 3 (𝑏 ∈ V → ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑))
54elv 3485 . 2 ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑)
6 csbopg 4891 . . . . 5 (𝑏 ∈ V → 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩)
76elv 3485 . . . 4 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦
8 vex 3484 . . . . . 6 𝑏 ∈ V
98csbconstgi 3920 . . . . 5 𝑏 / 𝑦𝑎 = 𝑎
108csbvargi 4435 . . . . 5 𝑏 / 𝑦𝑦 = 𝑏
119, 10opeq12i 4878 . . . 4 𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩ = ⟨𝑎, 𝑏
127, 11eqtri 2765 . . 3 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑎, 𝑏
13 dfsbcq 3790 . . 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 1540  Vcvv 3480  [wsbc 3788  csb 3899  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  reuop  6313
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