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Theorem bj-vjust2 33344
Description: Justification theorem for bj-df-v 33345. See also vjust 3403 and bj-vjust 33119. (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 2804 . . 3 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤)
2 bj-sbfvv 33100 . . . 4 ([𝑧 / 𝑦]⊤ ↔ ⊤)
3 df-clab 2804 . . . 4 (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤)
4 bj-sbfvv 33100 . . . 4 ([𝑧 / 𝑥]⊤ ↔ ⊤)
52, 3, 43bitr4ri 295 . . 3 ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
61, 5bitri 266 . 2 (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
76eqriv 2814 1 {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wtru 1638  [wsb 2061  wcel 2157  {cab 2803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-12 2215  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-sb 2062  df-clab 2804  df-cleq 2810
This theorem is referenced by: (None)
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