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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.) |
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 2069 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | spsbbi 2390 | . 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 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: sbcom3 2399 sbco3 2405 sbcom2 2433 sbal 2450 wl-equsb3 32516 wl-sbcom2d-lem1 32521 wl-sbcom3 32551 |
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