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| Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2176. (Revised by Wolf Lammen, 19-Jun-2023.) | 
| Ref | Expression | 
|---|---|
| nfceqi.1 | ⊢ 𝐴 = 𝐵 | 
| Ref | Expression | 
|---|---|
| nfceqi | ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfceqi.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eleq2i 2832 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) | 
| 3 | 2 | nfbii 1851 | . . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵) | 
| 4 | 3 | albii 1818 | . 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | 
| 5 | df-nfc 2891 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
| 6 | df-nfc 2891 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
| 7 | 4, 5, 6 | 3bitr4i 303 | 1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-cleq 2728 df-clel 2815 df-nfc 2891 | 
| This theorem is referenced by: nfcxfr 2902 nfcxfrd 2903 | 
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