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Mirrors > Home > ILE Home > Th. List > nfccdeq | GIF version |
Description: Variation of nfcdeq 2971 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 2323 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | equid 1711 | . . . . 5 ⊢ 𝑧 = 𝑧 | |
4 | 3 | cdeqth 2961 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧) |
5 | nfccdeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | 4, 5 | cdeqel 2970 | . . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
7 | 2, 6 | nfcdeq 2971 | . 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 7 | eqriv 2184 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2158 Ⅎwnfc 2316 CondEqwcdeq 2957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-cleq 2180 df-clel 2183 df-nfc 2318 df-cdeq 2958 |
This theorem is referenced by: (None) |
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