Metamath Proof Explorer
Description: Variant of nfan and commuted form of nfan1 . (Contributed by BTernaryTau, 31-Jul-2025)
|
|
Ref |
Expression |
|
Hypotheses |
nfan1c.1 |
⊢ Ⅎ 𝑥 𝜑 |
|
|
nfan1c.2 |
⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 ) |
|
Assertion |
nfan1c |
⊢ Ⅎ 𝑥 ( 𝜓 ∧ 𝜑 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
nfan1c.1 |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
nfan1c.2 |
⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 ) |
3 |
1 2
|
nfan1 |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 ) |
4 |
|
ancom |
⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
5 |
4
|
nfbii |
⊢ ( Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ Ⅎ 𝑥 ( 𝜓 ∧ 𝜑 ) ) |
6 |
3 5
|
mpbi |
⊢ Ⅎ 𝑥 ( 𝜓 ∧ 𝜑 ) |