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Mirrors > Home > MPE Home > Th. List > pwjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-pw 4296. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
pwjust | ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3759 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴)) | |
2 | 1 | cbvabv 2877 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑧 ∣ 𝑧 ⊆ 𝐴} |
3 | sseq1 3759 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
4 | 3 | cbvabv 2877 | . 2 ⊢ {𝑧 ∣ 𝑧 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} |
5 | 2, 4 | eqtri 2774 | 1 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1624 {cab 2738 ⊆ wss 3707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-clab 2739 df-cleq 2745 df-clel 2748 df-in 3714 df-ss 3721 |
This theorem is referenced by: (None) |
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