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

Theorem cbvabw 2891
Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2893 with a disjoint variable condition, which does not require ax-10 2145, ax-13 2391. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 23-May-2024.)
Hypotheses
Ref Expression
cbvabw.1 𝑦𝜑
cbvabw.2 𝑥𝜓
cbvabw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvabw {𝑥𝜑} = {𝑦𝜓}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvabw
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . . . . 8 𝑦 𝑥 = 𝑤
2 cbvabw.1 . . . . . . . 8 𝑦𝜑
31, 2nfim 1897 . . . . . . 7 𝑦(𝑥 = 𝑤𝜑)
4 nfv 1915 . . . . . . . 8 𝑥 𝑦 = 𝑤
5 cbvabw.2 . . . . . . . 8 𝑥𝜓
64, 5nfim 1897 . . . . . . 7 𝑥(𝑦 = 𝑤𝜓)
7 equequ1 2032 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑤𝑦 = 𝑤))
8 cbvabw.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
97, 8imbi12d 348 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 = 𝑤𝜑) ↔ (𝑦 = 𝑤𝜓)))
103, 6, 9cbvalv1 2362 . . . . . 6 (∀𝑥(𝑥 = 𝑤𝜑) ↔ ∀𝑦(𝑦 = 𝑤𝜓))
1110imbi2i 339 . . . . 5 ((𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤𝜑)) ↔ (𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓)))
1211albii 1821 . . . 4 (∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤𝜑)) ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓)))
13 df-sb 2070 . . . 4 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑥(𝑥 = 𝑤𝜑)))
14 df-sb 2070 . . . 4 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑤(𝑤 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜓)))
1512, 13, 143bitr4i 306 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
16 df-clab 2801 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
17 df-clab 2801 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
1815, 16, 173bitr4i 306 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1918eqriv 2819 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wnf 1785  [wsb 2069  wcel 2114  {cab 2800
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 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815
This theorem is referenced by:  cbvrabw  3465  cbvsbcw  3779  cbvrabcsfw  3896  rabsnifsb  4632  dfdmf  5742  dfrnf  5797  funfv2f  6734  abrexex2g  7651  bnj873  32270  ptrest  35014  poimirlem26  35041  poimirlem27  35042
  Copyright terms: Public domain W3C validator