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Mirrors > Home > MPE Home > Th. List > sbtrt | Structured version Visualization version GIF version |
Description: Partially closed form of sbtr 2524. Usage of this theorem is discouraged because it depends on ax-13 2380. (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 2516 | . 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 1535 Ⅎwnf 1781 [wsb 2064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2178 ax-13 2380 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ex 1778 df-nf 1782 df-sb 2065 |
This theorem is referenced by: sbtr 2524 |
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