MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcop Structured version   Visualization version   GIF version

Theorem sbcop 5500
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 5499 . . 3 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
32sbcbii 3852 . 2 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑)
4 sbcnestgw 4429 . . 3 (𝑏 ∈ V → ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑))
54elv 3483 . 2 ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑)
6 csbopg 4896 . . . . 5 (𝑏 ∈ V → 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩)
76elv 3483 . . . 4 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦
8 vex 3482 . . . . . 6 𝑏 ∈ V
98csbconstgi 3930 . . . . 5 𝑏 / 𝑦𝑎 = 𝑎
108csbvargi 4441 . . . . 5 𝑏 / 𝑦𝑦 = 𝑏
119, 10opeq12i 4883 . . . 4 𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩ = ⟨𝑎, 𝑏
127, 11eqtri 2763 . . 3 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑎, 𝑏
13 dfsbcq 3793 . . 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 3478  [wsbc 3791  csb 3908  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  reuop  6315
  Copyright terms: Public domain W3C validator