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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbnf | Structured version Visualization version GIF version |
Description: Move nonfree predicate in and out of substitution; see sbal 2165 and sbex 2287. (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-sbnf | ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2310 | . . . 4 ⊢ ([𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑)) | |
2 | sbal 2165 | . . . . 5 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
3 | 2 | imbi2i 338 | . . . 4 ⊢ (([𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)) |
4 | 1, 3 | bitri 277 | . . 3 ⊢ ([𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)) |
5 | 4 | albii 1819 | . 2 ⊢ (∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑) ↔ ∀𝑥([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)) |
6 | nf5 2289 | . . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
7 | 6 | sbbii 2080 | . . 3 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥(𝜑 → ∀𝑥𝜑)) |
8 | sbal 2165 | . . 3 ⊢ ([𝑧 / 𝑦]∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑)) | |
9 | 7, 8 | bitri 277 | . 2 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦](𝜑 → ∀𝑥𝜑)) |
10 | nf5 2289 | . 2 ⊢ (Ⅎ𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)) | |
11 | 5, 9, 10 | 3bitr4i 305 | 1 ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 Ⅎwnf 1783 [wsb 2068 |
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 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: bj-nfcf 34246 |
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