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Mirrors > Home > MPE Home > Th. List > nfsbvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfsbv 2328 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 2072 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
2 | nfv 1921 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝑦 | |
3 | nfv 1921 | . . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝑤 | |
4 | nfsbv.nf | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
5 | 3, 4 | nfim 1903 | . . . . 5 ⊢ Ⅎ𝑧(𝑥 = 𝑤 → 𝜑) |
6 | 5 | nfal 2321 | . . . 4 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑤 → 𝜑) |
7 | 2, 6 | nfim 1903 | . . 3 ⊢ Ⅎ𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
8 | 7 | nfal 2321 | . 2 ⊢ Ⅎ𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
9 | 1, 8 | nfxfr 1859 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1790 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-nf 1791 df-sb 2072 |
This theorem is referenced by: (None) |
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