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Mirrors > Home > MPE Home > Th. List > sbbid | Structured version Visualization version GIF version |
Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2130 and ax-13 2367. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2061. (Revised by Steven Nguyen, 11-Jul-2023.) |
Ref | Expression |
---|---|
sbbid.1 | ⊢ Ⅎ𝑥𝜑 |
sbbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbbid | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbbid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | sbbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimi 2202 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | spsbbi 2069 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: 2sbbid 2235 sbcom3 2501 sbco3 2508 wl-sbcom2d-lem1 37026 wl-2spsbbi 37032 wl-clabt 37065 |
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