Theorem bj-vjust2 33339

Description: Justification theorem for bj-df-v 33340. See also vjust 3341 and bj-vjust 33114. (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 2747	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤)
2		bj-sbfvv 33093	. . . 4 ⊢ ([𝑧 / 𝑦]⊤ ↔ ⊤)
3		df-clab 2747	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤)
4		bj-sbfvv 33093	. . . 4 ⊢ ([𝑧 / 𝑥]⊤ ↔ ⊤)
5	2, 3, 4	3bitr4ri 293	. . 3 ⊢ ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
6	1, 5	bitri 264	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤})
7	6	eqriv 2757	1 ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Syntax hints:   = wceq 1632  ⊤wtru 1633  [wsb 2046   ∈ wcel 2139  {cab 2746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753

This theorem is referenced by: (None)