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| Description: Variation of nfcdeq 3783 for classes. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2157. (Revised by GG, 19-May-2023.) (New usage is discouraged.) | 
| 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 2897 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | 
| 3 | eqid 2737 | . . . . 5 ⊢ 𝑧 = 𝑧 | |
| 4 | 3 | cdeqth 3773 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧) | 
| 5 | nfccdeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 6 | 4, 5 | cdeqel 3782 | . . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) | 
| 7 | 2, 6 | nfcdeq 3783 | . 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) | 
| 8 | 7 | eqriv 2734 | 1 ⊢ 𝐴 = 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 CondEqwcdeq 3769 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-cleq 2729 df-clel 2816 df-nfc 2892 df-cdeq 3770 | 
| This theorem is referenced by: (None) | 
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