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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 2140 and ax-13 2376. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2064. (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 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) | 
| 4 | spsbbi 2072 | . 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 206 ∀wal 1537 Ⅎwnf 1782 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: 2sbbid 2246 sbcom3 2510 sbco3 2517 wl-sbcom2d-lem1 37561 wl-2spsbbi 37567 wl-clabt 37600 | 
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