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Mirrors > Home > MPE Home > Th. List > sb6x | Structured version Visualization version GIF version |
Description: Equivalence involving substitution for a variable not free. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sb6x.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sb6x | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6x.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | sbf 2261 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
3 | biidd 263 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
4 | 1, 3 | equsal 2430 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) |
5 | 2, 4 | bitr4i 279 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 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-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: (None) |
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