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| Mirrors > Home > MPE Home > Th. List > sbtrt | Structured version Visualization version GIF version | ||
| Description: Partially closed form of sbtr 2521. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by BJ, 4-Jun-2019.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sbtrt.nf | ⊢ Ⅎ𝑦𝜑 |
| Ref | Expression |
|---|---|
| sbtrt | ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | stdpc4 2068 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑) | |
| 2 | sbtrt.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | 2 | sbid2 2513 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
| 4 | 1, 3 | sylib 218 | 1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1783 [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-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: sbtr 2521 |
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