Theorem pwjust 4581

Description: Soundness justification theorem for df-pw 4582. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
pwjust	⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem pwjust

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3989	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
2	1	cbvabv 2804	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑧 ∣ 𝑧 ⊆ 𝐴}
3		sseq1 3989	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
4	3	cbvabv 2804	. 2 ⊢ {𝑧 ∣ 𝑧 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}
5	2, 4	eqtri 2757	1 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}

Syntax hints:   = wceq 1539  {cab 2712   ⊆ wss 3931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-ss 3948

This theorem is referenced by: (None)