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| Mirrors > Home > ILE Home > Th. List > sbn | GIF version | ||
| Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
| Ref | Expression |
|---|---|
| sbn | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbnv 1912 | . . . 4 ⊢ ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑) | |
| 2 | 1 | sbbii 1788 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑) |
| 3 | sbnv 1912 | . . 3 ⊢ ([𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) | |
| 4 | 2, 3 | bitri 184 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
| 5 | ax-17 1549 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
| 6 | 5 | hbn 1677 | . . 3 ⊢ (¬ 𝜑 → ∀𝑧 ¬ 𝜑) |
| 7 | 6 | sbco2vh 1973 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑) |
| 8 | 5 | sbco2vh 1973 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
| 9 | 8 | notbii 670 | . 2 ⊢ (¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
| 10 | 4, 7, 9 | 3bitr3i 210 | 1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 ↔ wb 105 [wsb 1785 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 |
| This theorem is referenced by: sbcng 3039 difab 3442 rabeq0 3490 abeq0 3491 ssfirab 7033 |
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