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| Description: This is the closest we can get to df-sbc 3789 if we start from dfsbcq 3790 (see its comments) and dfsbcq2 3791. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| sbc8g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfsbcq 3790 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 2 | eleq1 2829 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
| 3 | df-clab 2715 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 4 | equid 2011 | . . . 4 ⊢ 𝑦 = 𝑦 | |
| 5 | dfsbcq2 3791 | . . . 4 ⊢ (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | 
| 7 | 3, 6 | bitr2i 276 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | 
| 8 | 1, 2, 7 | vtoclbg 3557 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 [wsb 2064 ∈ wcel 2108 {cab 2714 [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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 | 
| This theorem is referenced by: bnj984 34966 rusbcALT 44458 | 
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