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Mirrors > Home > MPE Home > Th. List > sbco2d | Structured version Visualization version GIF version |
Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) |
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 2198 | . . . 4 ⊢ Ⅎ𝑧(𝜑 → 𝜓) |
4 | 3 | sbco2 2552 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓)) |
5 | sbco2d.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
6 | 5 | sbrim 2312 | . . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓)) |
7 | 6 | sbbii 2080 | . . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓)) |
8 | 1 | sbrim 2312 | . . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
9 | 7, 8 | bitri 277 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓)) |
10 | 5 | sbrim 2312 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
11 | 4, 9, 10 | 3bitr3i 303 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
12 | 11 | pm5.74ri 274 | 1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1783 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: sbco3 2554 wl-clelsb3df 34867 |
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