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

Theorem sbcop 5482
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 5481 . . 3 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
32sbcbii 3833 . 2 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑)
4 sbcnestgw 4416 . . 3 (𝑏 ∈ V → ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑))
54elv 3479 . 2 ([𝑏 / 𝑦][𝑎, 𝑦⟩ / 𝑧]𝜑[𝑏 / 𝑦𝑎, 𝑦⟩ / 𝑧]𝜑)
6 csbopg 4884 . . . . 5 (𝑏 ∈ V → 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩)
76elv 3479 . . . 4 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦
8 vex 3477 . . . . . 6 𝑏 ∈ V
98csbconstgi 3911 . . . . 5 𝑏 / 𝑦𝑎 = 𝑎
108csbvargi 4428 . . . . 5 𝑏 / 𝑦𝑦 = 𝑏
119, 10opeq12i 4871 . . . 4 𝑏 / 𝑦𝑎, 𝑏 / 𝑦𝑦⟩ = ⟨𝑎, 𝑏
127, 11eqtri 2759 . . 3 𝑏 / 𝑦𝑎, 𝑦⟩ = ⟨𝑎, 𝑏
13 dfsbcq 3775 . . 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 296 1 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[𝑎, 𝑏⟩ / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  Vcvv 3473  [wsbc 3773  csb 3889  cop 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629
This theorem is referenced by:  reuop  6281
  Copyright terms: Public domain W3C validator