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Theorem bj-vjust2 32048
Description: Justification theorem for bj-df-v 32049. See also vjust 3078 and bj-vjust 31826. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-vjust2 {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Proof of Theorem bj-vjust2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2501 . . 3 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤)
2 bj-sbfvv 31802 . . . 4 ([𝑧 / 𝑦]⊤ ↔ ⊤)
3 df-clab 2501 . . . 4 (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤)
4 bj-sbfvv 31802 . . . 4 ([𝑧 / 𝑥]⊤ ↔ ⊤)
52, 3, 43bitr4ri 291 . . 3 ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
61, 5bitri 262 . 2 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
76eqriv 2511 1 {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wtru 1475  [wsb 1830  wcel 1938  {cab 2500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-12 1983  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1831  df-clab 2501  df-cleq 2507
This theorem is referenced by: (None)
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