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Mirrors > Home > MPE Home > Th. List > sb5 | Structured version Visualization version GIF version |
Description: Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2086 and even needs no disjoint variable condition, see sb1 2496. Theorem sb5f 2531 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2268. (Revised by Wolf Lammen, 4-Sep-2023.) |
Ref | Expression |
---|---|
sb5 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 2264 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | sbequ12 2243 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | equsexv 2259 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ [𝑦 / 𝑥]𝜑) |
4 | 3 | bicomi 225 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sb56 2268 2sb5 2273 sbnvOLD 2313 sb7f 2561 sbc2or 3778 sbc5 3797 |
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