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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvcllem | Structured version Visualization version GIF version |
Description: Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.) |
Ref | Expression |
---|---|
cbvcllem.y | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvcllem | ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvcllem.y | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cleq2lem 39961 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑋 ⊆ 𝑥 ∧ 𝜑) ↔ (𝑋 ⊆ 𝑦 ∧ 𝜓))) |
3 | 2 | cbvabv 2889 | 1 ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 {cab 2799 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 |
This theorem is referenced by: (None) |
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