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Mirrors > Home > MPE Home > Th. List > sbanOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sban 2086 as of 13-Aug-2023. Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbanOLD | ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbn 2287 | . . 3 ⊢ ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ [𝑦 / 𝑥](𝜑 → ¬ 𝜓)) | |
2 | sbim 2311 | . . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓)) | |
3 | sbn 2287 | . . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓) | |
4 | 3 | imbi2i 338 | . . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥] ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓)) |
5 | 2, 4 | bitri 277 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓)) |
6 | 1, 5 | xchbinx 336 | . 2 ⊢ ([𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓)) |
7 | df-an 399 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) | |
8 | 7 | sbbii 2081 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑦 / 𝑥] ¬ (𝜑 → ¬ 𝜓)) |
9 | df-an 399 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ¬ ([𝑦 / 𝑥]𝜑 → ¬ [𝑦 / 𝑥]𝜓)) | |
10 | 6, 8, 9 | 3bitr4i 305 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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