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Mirrors > Home > MPE Home > Th. List > sbtrt | Structured version Visualization version GIF version |
Description: Partially closed form of sbtr 2518. Usage of this theorem is discouraged because it depends on ax-13 2370. (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 2070 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑) | |
2 | sbtrt.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | sbid2 2510 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 1, 3 | sylib 217 | 1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1784 [wsb 2066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1781 df-nf 1785 df-sb 2067 |
This theorem is referenced by: sbtr 2518 |
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