Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sb6rft | Structured version Visualization version GIF version |
Description: A specialization of wl-equsal1t 34783. Closed form of sb6rf 2491. (Contributed by Wolf Lammen, 27-Jul-2019.) |
Ref | Expression |
---|---|
wl-sb6rft | ⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnf1 2158 | . . 3 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | |
2 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | sbequ12r 2254 | . . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)) | |
4 | 3 | a1i 11 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))) |
5 | 1, 2, 4 | wl-equsald 34781 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑)) |
6 | 5 | bicomd 225 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 Ⅎwnf 1784 [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-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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