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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 2192 and ax-13 2389. (Revised by Wolf Lammen, 24-Nov-2022.) |
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 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimpd 221 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
4 | 1, 3 | sbimd 2283 | . 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) |
5 | 2 | biimprd 240 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) |
6 | 1, 5 | sbimd 2283 | . 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)) |
7 | 4, 6 | impbid 204 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 Ⅎwnf 1882 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-nf 1883 df-sb 2068 |
This theorem is referenced by: sbcom3 2542 sbco3 2549 sbcom2 2578 sbcom2OLD 2579 sbal 2596 wl-equsb3 33881 wl-sbcom2d-lem1 33885 |
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