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| Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by GG, 12-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| sbc2ie.1 | ⊢ 𝐴 ∈ V | 
| sbc2ie.2 | ⊢ 𝐵 ∈ V | 
| sbc2ie.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbc2ie | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbc2ie.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | sbc2ie.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 ∈ V) | 
| 4 | sbc2ie.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 5 | 3, 4 | sbcied 3831 | . 2 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) | 
| 6 | 1, 5 | sbcie 3829 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 [wsbc 3787 | 
| 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-sbc 3788 | 
| This theorem is referenced by: sbc3ie 3867 brfi1uzind 14548 opfi1uzind 14551 wrd2ind 14762 isprs 18343 isdrs 18348 istos 18464 issrg 20186 isslmd 33209 rexrabdioph 42810 rmydioph 43031 rmxdioph 43033 expdiophlem2 43039 2reu8i 47130 | 
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