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Mirrors > Home > MPE Home > Th. List > sbor | Structured version Visualization version GIF version |
Description: Disjunction inside and outside of a substitution are equivalent. (Contributed by NM, 29-Sep-2002.) |
Ref | Expression |
---|---|
sbor | ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2304 | . . 3 ⊢ ([𝑦 / 𝑥](¬ 𝜑 → 𝜓) ↔ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbn 2281 | . . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
3 | 2 | imbi1i 350 | . . 3 ⊢ (([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥]𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
4 | 1, 3 | bitri 274 | . 2 ⊢ ([𝑦 / 𝑥](¬ 𝜑 → 𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
5 | df-or 845 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
6 | 5 | sbbii 2083 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ [𝑦 / 𝑥](¬ 𝜑 → 𝜓)) |
7 | df-or 845 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
8 | 4, 6, 7 | 3bitr4i 303 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 844 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-10 2141 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1787 df-nf 1791 df-sb 2072 |
This theorem is referenced by: sbcor 3773 unab 4238 |
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