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Theorem vjust 2558
 Description: Soundness justification theorem for df-v 2559. (Contributed by Rodolfo Medina, 27-Apr-2010.)
Assertion
Ref Expression
vjust {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Proof of Theorem vjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equid 1589 . . . . 5 𝑥 = 𝑥
21sbt 1667 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 1589 . . . . 5 𝑦 = 𝑦
43sbt 1667 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 163 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2027 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2027 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 201 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2037 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  [wsb 1645  {cab 2026 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033 This theorem is referenced by: (None)
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