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| Description: Class version of one implication of equvelv 2029. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| sbceqal | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqeq1 2740 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴)) | |
| 2 | eqeq1 2740 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
| 3 | 1, 2 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 → 𝐴 = 𝐵))) | 
| 4 | eqid 2736 | . . . 4 ⊢ 𝐴 = 𝐴 | |
| 5 | 4 | a1bi 362 | . . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 = 𝐴 → 𝐴 = 𝐵)) | 
| 6 | 3, 5 | bitr4di 289 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵)) | 
| 7 | 6 | spcgv 3595 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2107 | 
| 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-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 | 
| This theorem is referenced by: sbeqalb 3852 | 
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