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Mirrors > Home > MPE Home > Th. List > sbtrt | Structured version Visualization version GIF version |
Description: Partially closed form of sbtr 2557. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 2072 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑) | |
2 | sbtrt.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | sbid2 2549 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 1, 3 | sylib 220 | 1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 Ⅎwnf 1783 [wsb 2068 |
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 1969 ax-7 2014 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: sbtr 2557 |
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