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| Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| sbcied.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| sbcied.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | 
| sbciedf.3 | ⊢ Ⅎ𝑥𝜑 | 
| sbciedf.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) | 
| Ref | Expression | 
|---|---|
| sbciedf | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcied.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | sbciedf.4 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
| 3 | sbciedf.3 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 4 | sbcied.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
| 5 | 4 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) | 
| 6 | 3, 5 | alrimi 2213 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) | 
| 7 | sbciegft 3825 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | |
| 8 | 1, 2, 6, 7 | syl3anc 1373 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 [wsbc 3788 | 
| 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-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 | 
| This theorem is referenced by: sbc2iegf 3865 csbiebt 3928 sbcnestgfw 4421 sbcnestgf 4426 ovmpodxf 7583 sbc2iedf 32484 reuf1odnf 47119 ovmpordxf 48255 | 
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