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Mirrors > Home > MPE Home > Th. List > sbalOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbal 2166 as of 13-Aug-2023. Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbalOLD | ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfae 2455 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧 | |
2 | axc16gb 2263 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝜑 ↔ ∀𝑥𝜑)) | |
3 | 1, 2 | sbbid 2246 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑)) |
4 | axc16gb 2263 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | |
5 | 3, 4 | bitr3d 283 | . 2 ⊢ (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) |
6 | sbal1 2572 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | |
7 | 5, 6 | pm2.61i 184 | 1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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