Theorem vjust 3432

Description: Justification theorem for df-v 3433. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Assertion

Ref	Expression
vjust	⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Proof of Theorem vjust

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 2019	. . . 4 ⊢ 𝑥 = 𝑥
2	1	vexw 2723	. . 3 ⊢ 𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥}
3		equid 2019	. . . 4 ⊢ 𝑦 = 𝑦
4	3	vexw 2723	. . 3 ⊢ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦}
5	2, 4	2th 265	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦})
6	5	eqriv 2736	1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Syntax hints:   = wceq 1547   ∈ wcel 2119  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731

This theorem is referenced by: (None)