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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb8mot | Structured version Visualization version GIF version | ||
| Description: Substitution of variable in universal quantifier. Closed form of sb8mo 2600. (Contributed by Wolf Lammen, 11-Aug-2019.) | 
| Ref | Expression | 
|---|---|
| wl-sb8mot | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wl-sb8et 37555 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)) | |
| 2 | wl-sb8eut 37580 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)) | |
| 3 | 1, 2 | imbi12d 344 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))) | 
| 4 | moeu 2582 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
| 5 | moeu 2582 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 ∃wex 1778 Ⅎwnf 1782 [wsb 2063 ∃*wmo 2537 ∃!weu 2567 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: (None) | 
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