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Mirrors > Home > ILE Home > Th. List > sbco2d | GIF version |
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbco2d.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
sbco2d.2 | ⊢ (𝜑 → ∀𝑧𝜑) |
sbco2d.3 | ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) |
Ref | Expression |
---|---|
sbco2d | ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2d.2 | . . . . 5 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | sbco2d.3 | . . . . 5 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓)) | |
3 | 1, 2 | hbim1 1549 | . . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓)) |
4 | 3 | sbco2h 1935 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓)) |
5 | sbco2d.1 | . . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑) | |
6 | 5 | sbrim 1927 | . . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓)) |
7 | 6 | sbbii 1738 | . . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓)) |
8 | 1 | sbrim 1927 | . . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
9 | 7, 8 | bitri 183 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
10 | 5 | sbrim 1927 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
11 | 4, 9, 10 | 3bitr3i 209 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
12 | 11 | pm5.74ri 180 | 1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 [wsb 1735 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: (None) |
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