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Mirrors > Home > MPE Home > Th. List > sbco2d | Structured version Visualization version GIF version |
Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
sbco2d.1 | ⊢ Ⅎ𝑥𝜑 |
sbco2d.2 | ⊢ Ⅎ𝑧𝜑 |
sbco2d.3 | ⊢ (𝜑 → Ⅎ𝑧𝜓) |
Ref | Expression |
---|---|
sbco2d | ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2d.2 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
2 | sbco2d.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
3 | 1, 2 | nfim1 2164 | . . . 4 ⊢ Ⅎ𝑧(𝜑 → 𝜓) |
4 | 3 | sbco2 2507 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓)) |
5 | sbco2d.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
6 | 5 | sbrim 2279 | . . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓)) |
7 | 6 | sbbii 2054 | . . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓)) |
8 | 1 | sbrim 2279 | . . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
9 | 7, 8 | bitri 276 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
10 | 5 | sbrim 2279 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
11 | 4, 9, 10 | 3bitr3i 302 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
12 | 11 | pm5.74ri 273 | 1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1765 [wsb 2042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 |
This theorem is referenced by: sbco3 2509 wl-clelsb3df 34394 |
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