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Theorem bj-vjust 34246
 Description: Justification theorem for bj-df-v 34247. See also vjust 3501. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-vjust {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Proof of Theorem bj-vjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2805 . . 3 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤)
2 sbv 2091 . . . 4 ([𝑧 / 𝑦]⊤ ↔ ⊤)
3 df-clab 2805 . . . 4 (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤)
4 sbv 2091 . . . 4 ([𝑧 / 𝑥]⊤ ↔ ⊤)
52, 3, 43bitr4ri 305 . . 3 ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
61, 5bitri 276 . 2 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
76eqriv 2823 1 {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530  ⊤wtru 1531  [wsb 2062   ∈ wcel 2107  {cab 2804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-clab 2805  df-cleq 2819 This theorem is referenced by: (None)
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