Theorem bj-vjust2 33015

Description: Justification theorem for bj-df-v 33016. See also vjust 3201 and bj-vjust 32786. (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 2609	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤)
2		bj-sbfvv 32765	. . . 4 ⊢ ([𝑧 / 𝑦]⊤ ↔ ⊤)
3		df-clab 2609	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤)
4		bj-sbfvv 32765	. . . 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 2619	1 ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Syntax hints:   = wceq 1483  ⊤wtru 1484  [wsb 1880   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615

This theorem is referenced by: (None)