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| Mirrors > Home > MPE Home > Th. List > csbie2 | Structured version Visualization version GIF version | ||
| Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.) |
| Ref | Expression |
|---|---|
| csbie2t.1 | ⊢ 𝐴 ∈ V |
| csbie2t.2 | ⊢ 𝐵 ∈ V |
| csbie2.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| csbie2 | ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbie2.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
| 2 | 1 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
| 3 | csbie2t.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 4 | csbie2t.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 5 | 3, 4 | csbie2t 3937 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
| 6 | 2, 5 | ax-mp 5 | 1 ⊢ ⦋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: fsumcnv 15809 fprodcnv 16019 rnghmval 20440 dfrhm2 20474 mamufval 22396 mvmulfval 22548 vtxdgfval 29485 grtri 47907 |
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