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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 2377. Check out nfs1v 2156 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 2195 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
| 3 | 2 | hbsb3 2492 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
| 4 | 3 | nf5i 2146 | 1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: 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: sb8 2522 sb8e 2523 |
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