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| Mirrors > Home > MPE Home > Th. List > sbco3 | Structured version Visualization version GIF version | ||
| Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sbco3 | ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | drsb1 2500 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)) | |
| 2 | nfae 2438 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
| 3 | sbequ12a 2254 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | |
| 4 | 3 | sps 2185 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) |
| 5 | 2, 4 | sbbid 2246 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)) |
| 6 | 1, 5 | bitr3d 281 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)) |
| 7 | nfnae 2439 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 8 | nfnae 2439 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 9 | nfsb2 2488 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
| 10 | 7, 8, 9 | sbco2d 2517 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)) |
| 11 | sbco 2512 | . . . 4 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑) | |
| 12 | 11 | sbbii 2076 | . . 3 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
| 13 | 10, 12 | bitr3di 286 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)) |
| 14 | 6, 13 | pm2.61i 182 | 1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 [wsb 2064 |
| 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 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: sbcom 2519 |
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