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Mirrors > Home > MPE Home > Th. List > sb6f | Structured version Visualization version GIF version |
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2478 and does not require the nonfreeness hypothesis. Theorem sb6 2089 replaces the nonfreeness hypothesis with a disjoint variable condition on 𝑥, 𝑦 and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb6f.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb6f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6f.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nf5ri 2189 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | sbimi 2078 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑) |
4 | sb4a 2479 | . . 3 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | sb2 2478 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
7 | 5, 6 | impbii 208 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 Ⅎwnf 1786 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 |
This theorem is referenced by: sb5f 2497 bj-sbievv 35360 |
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