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Mirrors > Home > ILE Home > Th. List > nfceqi | GIF version |
Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfceqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
nfceqi | ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqi.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2231 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | 2 | nfbii 1460 | . . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵) |
4 | 3 | albii 1457 | . 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) |
5 | df-nfc 2295 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
6 | df-nfc 2295 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4i 211 | 1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1340 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 |
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 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-cleq 2157 df-clel 2160 df-nfc 2295 |
This theorem is referenced by: nfcxfr 2303 nfcxfrd 2304 |
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