Theorem pwjust 3506

Description: Soundness justification theorem for df-pw 3507. (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 3115	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
2	1	cbvabv 2262	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑧 ∣ 𝑧 ⊆ 𝐴}
3		sseq1 3115	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
4	3	cbvabv 2262	. 2 ⊢ {𝑧 ∣ 𝑧 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}
5	2, 4	eqtri 2158	1 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}

Syntax hints:   = wceq 1331  {cab 2123   ⊆ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079

This theorem is referenced by: (None)