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| Mirrors > Home > MPE Home > Th. List > csbie2t | Structured version Visualization version GIF version | ||
| Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3938). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.) |
| Ref | Expression |
|---|---|
| csbie2t.1 | ⊢ 𝐴 ∈ V |
| csbie2t.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| csbie2t | ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
| 2 | nfcvd 2906 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → Ⅎ𝑥𝐷) | |
| 3 | csbie2t.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 4 | 3 | a1i 11 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V) |
| 5 | nfa2 2176 | . . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
| 6 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
| 7 | 5, 6 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) |
| 8 | nfcvd 2906 | . . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝐷) | |
| 9 | csbie2t.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 10 | 9 | a1i 11 | . . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
| 11 | 2sp 2186 | . . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)) | |
| 12 | 11 | impl 455 | . . 3 ⊢ (((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
| 13 | 7, 8, 10, 12 | csbiedf 3929 | . 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
| 14 | 1, 2, 4, 13 | csbiedf 3929 | 1 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: csbie2 3938 |
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