![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbbi | Structured version Visualization version GIF version |
Description: Equivalence inside and outside of a substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbbiv 2341. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
sbbi | ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi2 468 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
2 | 1 | sbbii 2076 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ [𝑦 / 𝑥]((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
3 | sbim 2526 | . . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
4 | sbim 2526 | . . . 4 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜑) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)) | |
5 | 3, 4 | anbi12i 622 | . . 3 ⊢ (([𝑦 / 𝑥](𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜓 → 𝜑)) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))) |
6 | sban 2530 | . . 3 ⊢ ([𝑦 / 𝑥]((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ([𝑦 / 𝑥](𝜑 → 𝜓) ∧ [𝑦 / 𝑥](𝜓 → 𝜑))) | |
7 | dfbi2 468 | . . 3 ⊢ (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))) | |
8 | 5, 6, 7 | 3bitr4i 295 | . 2 ⊢ ([𝑦 / 𝑥]((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
9 | 2, 8 | bitri 267 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-12 2222 ax-13 2391 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ex 1881 df-nf 1885 df-sb 2070 |
This theorem is referenced by: spsbbi 2533 sblbis 2535 sbrbis 2536 pm13.183 3563 sbcbig 3707 sb8iota 6093 bj-sbidmOLD 33355 |
Copyright terms: Public domain | W3C validator |