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Mirrors > Home > MPE Home > Th. List > sbco2 | Structured version Visualization version GIF version |
Description: A composition law for substitution. (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 2097 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)) | |
2 | sbequ 2364 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | bitr3d 269 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
4 | 3 | sps 2043 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
5 | nfnae 2306 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
6 | sbco2.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
7 | 6 | nfsb4 2378 | . . 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 2397 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
10 | 4, 9 | pm2.61i 175 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: sbco2d 2404 equsb3ALT 2421 elsb3 2422 elsb4 2423 sb7f 2441 sbco4lem 2453 sbco4 2454 eqsb3 2715 clelsb3 2716 cbvab 2733 sbralie 3160 sbcco 3425 clelsb3f 28704 bj-clelsb3 32042 |
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