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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 2096 and even needs no disjoint variable condition, see sb1 2479. Theorem sb5f 2502 replaces the disjoint variable condition with a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Revised by Wolf Lammen, 4-Sep-2023.) |
Ref | Expression |
---|---|
sb5 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2094 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sbalex 2243 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | bitr4i 281 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃wex 1786 [wsb 2073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-12 2178 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2074 |
This theorem is referenced by: sb56OLD 2277 2sb5 2279 sb7f 2530 clelab 2875 sbc2or 3691 sbc5ALT 3711 |
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