Theorem vjust 3497

Description: Soundness justification theorem for df-v 3498. (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	. . . . 5 ⊢ 𝑥 = 𝑥
2	1	sbt 2071	. . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥
3		equid 2019	. . . . 5 ⊢ 𝑦 = 𝑦
4	3	sbt 2071	. . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦
5	2, 4	2th 266	. . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6		df-clab 2802	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7		df-clab 2802	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
8	5, 6, 7	3bitr4i 305	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦})
9	8	eqriv 2820	1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Syntax hints:   = wceq 1537  [wsb 2069   ∈ wcel 2114  {cab 2801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816

This theorem is referenced by: (None)