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Mirrors > Home > MPE Home > Th. List > nfs1 | Structured version Visualization version GIF version |
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2367. Check out nfs1v 2146 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfs1.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
nfs1 | ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nf5ri 2184 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | hbsb3 2482 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
4 | 3 | nf5i 2135 | 1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: sb8 2512 sb8e 2513 |
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