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Mirrors > Home > ILE Home > Th. List > sbanv | GIF version |
Description: Version of sban 1870 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Ref | Expression |
---|---|
sbanv | ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1807 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓))) | |
2 | sb6 1807 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | sb6 1807 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) | |
4 | 2, 3 | anbi12i 447 | . . 3 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
5 | 19.26 1410 | . . 3 ⊢ (∀𝑥((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜓))) | |
6 | pm4.76 568 | . . . 4 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ (𝑥 = 𝑦 → (𝜑 ∧ 𝜓))) | |
7 | 6 | albii 1399 | . . 3 ⊢ (∀𝑥((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓))) |
8 | 4, 5, 7 | 3bitr2i 206 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓))) |
9 | 1, 8 | bitr4i 185 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-sb 1686 |
This theorem is referenced by: sban 1870 |
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