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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 3934). (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 2140 | . 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
2 | nfcvd 2899 | . 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 2165 | . . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
6 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
7 | 5, 6 | nfan 1894 | . . 3 ⊢ Ⅎ𝑦(∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) |
8 | nfcvd 2899 | . . 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 2174 | . . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)) | |
12 | 11 | impl 454 | . . 3 ⊢ (((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
13 | 7, 8, 10, 12 | csbiedf 3923 | . 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
14 | 1, 2, 4, 13 | csbiedf 3923 | 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 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⦋csb 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-v 3473 df-sbc 3777 df-csb 3893 |
This theorem is referenced by: csbie2 3934 |
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