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Theorem cbvab 2912
 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 2477 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 cbvab.2 . . . . . 6 𝑥𝜓
4 cbvab.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2468 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65sbbii 2027 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
72, 6bitr3i 269 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2760 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2760 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 295 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2776 1 {𝑥𝜑} = {𝑦𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507  Ⅎwnf 1746  [wsb 2015   ∈ wcel 2050  {cab 2759 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772 This theorem is referenced by:  cbvabvOLD  2913  cbvrab  3412  cdeqab1  3673  cbvsbc  3711  cbvrabcsf  3824  rabsnifsb  4532  dfdmf  5615  dfrnf  5663  funfv2f  6580  abrexex2g  7477  bnj873  31840  ptrest  34329  poimirlem26  34356  poimirlem27  34357
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