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Mirrors > Home > ILE Home > Th. List > nfccdeq | GIF version |
Description: Variation of nfcdeq 2901 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfccdeq.1 | ⊢ Ⅎ𝑥𝐴 |
nfccdeq.2 | ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nfccdeq | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfccdeq.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2273 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | equid 1677 | . . . . 5 ⊢ 𝑧 = 𝑧 | |
4 | 3 | cdeqth 2891 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧) |
5 | nfccdeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | 4, 5 | cdeqel 2900 | . . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
7 | 2, 6 | nfcdeq 2901 | . 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 7 | eqriv 2134 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Ⅎwnfc 2266 CondEqwcdeq 2887 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-cleq 2130 df-clel 2133 df-nfc 2268 df-cdeq 2888 |
This theorem is referenced by: (None) |
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