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

Theorem opelopabsbALT 5407
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 5408, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbALT (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem opelopabsbALT
StepHypRef Expression
1 excom 2159 . . 3 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3495 . . . . . . 7 𝑧 ∈ V
3 vex 3495 . . . . . . 7 𝑤 ∈ V
42, 3opth 5359 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥𝑤 = 𝑦))
5 equcom 2016 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
6 equcom 2016 . . . . . . 7 (𝑤 = 𝑦𝑦 = 𝑤)
75, 6anbi12ci 627 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑦 = 𝑤𝑥 = 𝑧))
84, 7bitri 276 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤𝑥 = 𝑧))
98anbi1i 623 . . . 4 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1092exbii 1840 . . 3 (∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
111, 10bitri 276 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
12 elopab 5405 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13 2sb5 2273 . 2 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1411, 12, 133bitr4i 304 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  [wsb 2060  wcel 2105  cop 4563  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120
This theorem is referenced by:  inopab  5694  cnvopab  5990  brabsb2  35878
  Copyright terms: Public domain W3C validator