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| Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2109 and df-clel 2815. (Revised by WL and SN, 23-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| nfceqdf.1 | ⊢ Ⅎ𝑥𝜑 | 
| nfceqdf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| nfceqdf | ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfceqdf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfceqdf.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | eleq2w2 2732 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | 
| 5 | 1, 4 | nfbidf 2223 | . . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)) | 
| 6 | 5 | albidv 1919 | . 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)) | 
| 7 | df-nfc 2891 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
| 8 | df-nfc 2891 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
| 9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ 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-9 2117 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-cleq 2728 df-nfc 2891 | 
| This theorem is referenced by: nfopd 4889 dfnfc2 4928 nfimad 6086 nffvd 6917 riotasv2d 38959 nfcxfrdf 38968 nfded 38969 nfded2 38970 nfunidALT2 38971 | 
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