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Mirrors > Home > MPE Home > Th. List > sbco2 | Structured version Visualization version GIF version |
Description: A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2343 and sbco2vv 2099. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) |
Ref | Expression |
---|---|
sbco2.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2 | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 2243 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)) | |
2 | sbequ 2081 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | bitr3d 282 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
4 | 3 | sps 2174 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
5 | nfnae 2448 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
6 | sbco2.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
7 | 6 | nfsb4 2533 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
8 | 2 | a1i 11 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
9 | 5, 7, 8 | sbied 2538 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
10 | 4, 9 | pm2.61i 183 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∀wal 1526 Ⅎwnf 1775 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sbco2d 2547 sb7f 2561 cbvab 2888 clelsb3f 2979 clelsb3fOLD 2980 sbcco 3795 |
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