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Mirrors > Home > MPE Home > Th. List > dfsb1 | Structured version Visualization version GIF version |
Description: Alternate definition of substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). This was the original definition before df-sb 2066. Note that it does not require dummy variables in its definiens; this is done by having 𝑥 free in the first conjunct and bound in the second. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by BJ, 9-Jul-2023.) Revise df-sb 2066. (Revised by Wolf Lammen, 29-Jul-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ2 2239 | . . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | |
2 | 1 | com12 32 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑)) |
3 | sb1 2475 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
4 | 2, 3 | jca 510 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
5 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | sbequ1 2238 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
7 | 5, 6 | embantd 59 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)) |
8 | 7 | sps 2176 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)) |
9 | 8 | adantrd 490 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑)) |
10 | sb3 2474 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) | |
11 | 10 | adantld 489 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑)) |
12 | 9, 11 | pm2.61i 182 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑) |
13 | 4, 12 | impbii 208 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1537 ∃wex 1779 [wsb 2065 |
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 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 ax-13 2369 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-nf 1784 df-sb 2066 |
This theorem is referenced by: drsb1 2492 bj-dfsb2 36021 frege55b 42952 |
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