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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 37914 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑋 ⊆ 𝑥 ∧ 𝜑) ↔ (𝑋 ⊆ 𝑦 ∧ 𝜓))) |
3 | 2 | cbvabv 2747 | 1 ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 {cab 2608 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 |
This theorem is referenced by: (None) |
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