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Mirrors > Home > ILE Home > Th. List > sbnv | GIF version |
Description: Version of sbn 1939 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.) |
Ref | Expression |
---|---|
sbnv | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1873 | . . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) | |
2 | alinexa 1590 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
4 | sb5 1874 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
5 | 3, 4 | xchbinxr 673 | 1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 ∃wex 1479 [wsb 1749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-sb 1750 |
This theorem is referenced by: sbn 1939 |
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