![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ax11-lem5 | Structured version Visualization version GIF version |
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
Ref | Expression |
---|---|
wl-ax11-lem5 | ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12r 2237 | . . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑 ↔ 𝜑)) | |
2 | 1 | sps 2171 | . 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑 ↔ 𝜑)) |
3 | 2 | dral1 2433 | 1 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2164 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: wl-ax11-lem6 36979 |
Copyright terms: Public domain | W3C validator |