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.

Step	Hyp	Ref	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]⊤ ↔ ⊤)
5	2, 3, 4	3bitr4ri 291	. . 3 ⊢ ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
6	1, 5	bitri 262	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
7	6	eqriv 2511	1 ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

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)