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Mirrors > Home > MPE Home > Th. List > cdeqab1 | Structured version Visualization version GIF version |
Description: Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqnot.1 | ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cdeqab1 | ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cdeqri 3421 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
3 | 2 | cbvabv 2747 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
4 | 3 | cdeqth 3422 | 1 ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 {cab 2608 CondEqwcdeq 3418 |
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-cdeq 3419 |
This theorem is referenced by: (None) |
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