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Mirrors > Home > ILE Home > Th. List > sborv | GIF version |
Description: Version of sbor 1828 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.) |
Ref | Expression |
---|---|
sborv | ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 1767 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ∨ 𝜓))) | |
2 | andi 731 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜓))) | |
3 | 2 | exbii 1496 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑥((𝑥 = 𝑦 ∧ 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜓))) |
4 | 19.43 1519 | . . 3 ⊢ (∃𝑥((𝑥 = 𝑦 ∧ 𝜑) ∨ (𝑥 = 𝑦 ∧ 𝜓)) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ∨ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
5 | 1, 3, 4 | 3bitri 195 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ∨ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
6 | sb5 1767 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | sb5 1767 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) | |
8 | 6, 7 | orbi12i 681 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ∨ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
9 | 5, 8 | bitr4i 176 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 629 ∃wex 1381 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: sbor 1828 |
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