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Mirrors > Home > MPE Home > Th. List > sb5f | Structured version Visualization version GIF version |
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2496 and does not require the non-freeness hypothesis. Theorem sb5 2267 replaces the non-freeness hypothesis with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sb6f.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb5f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6f.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb6f 2530 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | 1 | equs45f 2474 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 2, 3 | bitr4i 279 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 [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 ax-13 2381 |
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: sb7f 2561 |
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