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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 2340 and does not require the non-freeness hypothesis. Theorem sb6 2417 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
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 2053 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | sbimi 1873 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑) |
4 | sb4a 2345 | . . 3 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | sb2 2340 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
7 | 5, 6 | impbii 198 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: sb5f 2374 |
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