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Theorem cbvabw 2263
 Description: Version of cbvab 2264 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-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 variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvabw.1 . . . . 5 𝑦𝜑
21sbco2v 1922 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 cbvabw.2 . . . . . 6 𝑥𝜓
4 cbvabw.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbiev 1766 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65sbbii 1739 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
72, 6bitr3i 185 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2127 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2127 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 211 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2137 1 {𝑥𝜑} = {𝑦𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  [wsb 1736  {cab 2126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133 This theorem is referenced by:  cbvsbcw  2937
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