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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 2375. (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 2498 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)) | |
2 | nfae 2436 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
3 | sbequ12a 2252 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | |
4 | 3 | sps 2183 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) |
5 | 2, 4 | sbbid 2244 | . . 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 2437 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
8 | nfnae 2437 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
9 | nfsb2 2486 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
10 | 7, 8, 9 | sbco2d 2515 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)) |
11 | sbco 2510 | . . . 4 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑) | |
12 | 11 | sbbii 2074 | . . 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 1535 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: sbcom 2517 |
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