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Mirrors > Home > NFE Home > Th. List > csbie2t | GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3181). (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 1788 | . 2 ⊢ Ⅎx∀x∀y((x = A ∧ y = B) → C = D) | |
2 | nfcvd 2490 | . 2 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → ℲxD) | |
3 | csbie2t.1 | . . 3 ⊢ A ∈ V | |
4 | 3 | a1i 10 | . 2 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → A ∈ V) |
5 | nfa2 1855 | . . . 4 ⊢ Ⅎy∀x∀y((x = A ∧ y = B) → C = D) | |
6 | nfv 1619 | . . . 4 ⊢ Ⅎy x = A | |
7 | 5, 6 | nfan 1824 | . . 3 ⊢ Ⅎy(∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) |
8 | nfcvd 2490 | . . 3 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → ℲyD) | |
9 | csbie2t.2 | . . . 4 ⊢ B ∈ V | |
10 | 9 | a1i 10 | . . 3 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → B ∈ V) |
11 | sp 1747 | . . . . 5 ⊢ (∀y((x = A ∧ y = B) → C = D) → ((x = A ∧ y = B) → C = D)) | |
12 | 11 | sps 1754 | . . . 4 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → ((x = A ∧ y = B) → C = D)) |
13 | 12 | impl 603 | . . 3 ⊢ (((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) ∧ y = B) → C = D) |
14 | 7, 8, 10, 13 | csbiedf 3173 | . 2 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → [B / y]C = D) |
15 | 1, 2, 4, 14 | csbiedf 3173 | 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 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 Vcvv 2859 [csb 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: csbie2 3181 |
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