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Theorem cbvab 2888
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 2546 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 cbvab.2 . . . . . 6 𝑥𝜓
4 cbvab.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2537 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65sbbii 2072 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
72, 6bitr3i 278 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2797 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2797 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 304 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2815 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wnf 1775  [wsb 2060  wcel 2105  {cab 2796
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-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811
This theorem is referenced by:  cbvabvOLD  2889  cbvrab  3488  cdeqab1  3760  cbvsbc  3803  cbvrabcsf  3925
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