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Theorem cbvab 2895
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Hypotheses
Ref Expression
cbvab.1 𝑦𝜑
cbvab.2 𝑥𝜓
cbvab.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvab {𝑥𝜑} = {𝑦𝜓}

Proof of Theorem cbvab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvab.1 . . . . 5 𝑦𝜑
21sbco2 2562 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 cbvab.2 . . . . . 6 𝑥𝜓
4 cbvab.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2555 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65sbbii 2056 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
72, 6bitr3i 266 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2758 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2758 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 292 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2768 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wnf 1856  [wsb 2049  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764
This theorem is referenced by:  cbvabv  2896  cbvrab  3348  cbvsbc  3616  cbvrabcsf  3717  rabsnifsb  4393  dfdmf  5455  dfrnf  5502  funfv2f  6409  abrexex2g  7291  abrexex2OLD  7297  bnj873  31332  cnfinltrel  33578  ptrest  33741  poimirlem26  33768  poimirlem27  33769
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