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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 2092 and even needs no disjoint variable condition, see sb1 2492. Theorem sb5f 2516 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2274. (Revised by Wolf Lammen, 4-Sep-2023.) |
Ref | Expression |
---|---|
sb5 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 2157 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | sbequ12 2250 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | equsexv 2266 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ [𝑦 / 𝑥]𝜑) |
4 | 3 | bicomi 227 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: sb56 2274 2sb5 2278 sb7f 2545 sbc2or 3729 sbc5 3748 |
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