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 2424. (Contributed by Wolf Lammen, 27-Jul-2019.) |
Ref | Expression |
---|---|
wl-sb8t | ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2028 | . 2 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
2 | nfnf1 2031 | . . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑 | |
3 | 2 | nfal 2153 | . 2 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
4 | sp 2053 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑) | |
5 | wl-nfs1t 33324 | . . 3 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
6 | 5 | sps 2055 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
7 | sbequ12 2111 | . . 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 2270 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 Ⅎwnf 1708 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: wl-sb8et 33334 wl-sbhbt 33335 |
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