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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbc2iedf | Structured version Visualization version GIF version | ||
| Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| sbc2iedf.1 | ⊢ Ⅎ𝑥𝜑 | 
| sbc2iedf.2 | ⊢ Ⅎ𝑦𝜑 | 
| sbc2iedf.3 | ⊢ Ⅎ𝑥𝜒 | 
| sbc2iedf.4 | ⊢ Ⅎ𝑦𝜒 | 
| sbc2iedf.5 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| sbc2iedf.6 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| sbc2iedf.7 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| sbc2iedf | ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbc2iedf.5 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | sbc2iedf.6 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊) | 
| 4 | sbc2iedf.7 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
| 5 | 4 | anassrs 467 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) | 
| 6 | sbc2iedf.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 7 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
| 8 | 6, 7 | nfan 1898 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴) | 
| 9 | sbc2iedf.4 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
| 10 | 9 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜒) | 
| 11 | 3, 5, 8, 10 | sbciedf 3830 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| 12 | sbc2iedf.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 13 | sbc2iedf.3 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
| 14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | 
| 15 | 1, 11, 12, 14 | sbciedf 3830 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 [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-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: rspc2daf 32486 | 
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