![]() |
Mathbox for BTernaryTau |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfan1c | Structured version Visualization version GIF version |
Description: Variant of nfan 1898 and commuted form of nfan1 2196. (Contributed by BTernaryTau, 31-Jul-2025.) |
Ref | Expression |
---|---|
nfan1c.1 | ⊢ Ⅎ𝑥𝜑 |
nfan1c.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfan1c | ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfan1c.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfan1c.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 1, 2 | nfan1 2196 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
4 | ancom 460 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
5 | 4 | nfbii 1850 | . 2 ⊢ (Ⅎ𝑥(𝜑 ∧ 𝜓) ↔ Ⅎ𝑥(𝜓 ∧ 𝜑)) |
6 | 3, 5 | mpbi 230 | 1 ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-12 2173 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ex 1778 df-nf 1782 |
This theorem is referenced by: dvelimalcased 35043 dvelimexcased 35045 |
Copyright terms: Public domain | W3C validator |