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Mirrors > Home > MPE Home > Th. List > spsbbi | Structured version Visualization version GIF version |
Description: Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) |
Ref | Expression |
---|---|
spsbbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2353 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → [𝑦 / 𝑥](𝜑 ↔ 𝜓)) | |
2 | sbbi 2401 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | |
3 | 1, 2 | sylib 208 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: sbbid 2403 sbeqi 33968 |
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