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Mirrors > Home > ILE Home > Th. List > nfccdeq | GIF version |
Description: Variation of nfcdeq 2859 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 2234 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | equid 1645 | . . . . 5 ⊢ 𝑧 = 𝑧 | |
4 | 3 | cdeqth 2849 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧) |
5 | nfccdeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | 4, 5 | cdeqel 2858 | . . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
7 | 2, 6 | nfcdeq 2859 | . 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 7 | eqriv 2097 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∈ wcel 1448 Ⅎwnfc 2227 CondEqwcdeq 2845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-cleq 2093 df-clel 2096 df-nfc 2229 df-cdeq 2846 |
This theorem is referenced by: (None) |
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