| Mathbox for Wolf Lammen |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb8t | Structured version Visualization version GIF version | ||
| Description: Substitution of variable in universal quantifier. Closed form of sb8 2522. (Contributed by Wolf Lammen, 27-Jul-2019.) |
| Ref | Expression |
|---|---|
| wl-sb8t | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
| 2 | nfnf1 2154 | . . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑 | |
| 3 | 2 | nfal 2323 | . 2 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
| 4 | sp 2183 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑) | |
| 5 | wl-nfs1t 37538 | . . 3 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
| 6 | 5 | sps 2185 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
| 7 | sbequ12 2251 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 8 | 7 | a1i 11 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
| 9 | 1, 3, 4, 6, 8 | cbv2 2408 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀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 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: wl-sb8et 37554 wl-sbhbt 37555 |
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