Metamath Proof Explorer
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
bnj1096.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
|
|
bnj1096.2 |
⊢ ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) ) |
|
Assertion |
bnj1096 |
⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1096.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
2 |
|
bnj1096.2 |
⊢ ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) ) |
3 |
|
ax-5 |
⊢ ( 𝜒 → ∀ 𝑥 𝜒 ) |
4 |
|
ax-5 |
⊢ ( 𝜃 → ∀ 𝑥 𝜃 ) |
5 |
|
ax-5 |
⊢ ( 𝜏 → ∀ 𝑥 𝜏 ) |
6 |
3 4 5 1
|
bnj982 |
⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) → ∀ 𝑥 ( 𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑 ) ) |
7 |
2 6
|
hbxfrbi |
⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) |