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Mirrors > Home > ILE Home > Th. List > cdeqab | 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 |
---|---|
cdeqab | ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cdeqri 2923 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
3 | 2 | abbidv 2275 | . 2 ⊢ (𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) |
4 | 3 | cdeqi 2922 | 1 ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1335 {cab 2143 CondEqwcdeq 2920 |
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-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-cdeq 2921 |
This theorem is referenced by: (None) |
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