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| Mirrors > Home > MPE Home > Th. List > nfsbvOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of nfsbv 2330 as of 25-Oct-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfsbv.nf | ⊢ Ⅎ𝑧𝜑 |
| Ref | Expression |
|---|---|
| nfsbvOLD | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-sb 2065 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
| 2 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝑦 | |
| 3 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝑤 | |
| 4 | nfsbv.nf | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
| 5 | 3, 4 | nfim 1896 | . . . . 5 ⊢ Ⅎ𝑧(𝑥 = 𝑤 → 𝜑) |
| 6 | 5 | nfal 2323 | . . . 4 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑤 → 𝜑) |
| 7 | 2, 6 | nfim 1896 | . . 3 ⊢ Ⅎ𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
| 8 | 7 | nfal 2323 | . 2 ⊢ Ⅎ𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
| 9 | 1, 8 | nfxfr 1853 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 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-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: (None) |
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