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| Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| sbc3ie.1 | ⊢ 𝐴 ∈ V | 
| sbc3ie.2 | ⊢ 𝐵 ∈ V | 
| sbc3ie.3 | ⊢ 𝐶 ∈ V | 
| sbc3ie.4 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbc3ie | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbc3ie.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | sbc3ie.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | sbc3ie.3 | . . . 4 ⊢ 𝐶 ∈ V | |
| 4 | 3 | a1i 11 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 ∈ V) | 
| 5 | sbc3ie.4 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
| 6 | 5 | 3expa 1118 | . . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | 
| 7 | 4, 6 | sbcied 3831 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑 ↔ 𝜓)) | 
| 8 | 1, 2, 7 | sbc2ie 3865 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = 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-3an 1088 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: isdlat 18568 islmod 20863 isslmd 33209 hdmap1fval 41799 hdmapfval 41830 hgmapfval 41889 rmydioph 43031 | 
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