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Theorem vjust 3174
 Description: Soundness justification theorem for df-v 3175. (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 1926 . . . . 5 𝑥 = 𝑥
21sbt 2407 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 1926 . . . . 5 𝑦 = 𝑦
43sbt 2407 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 253 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2597 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2597 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 291 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2607 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603 This theorem is referenced by: (None)
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