Metamath Proof Explorer
Description: Weak version of alcom and biconditional form of alcomiw . Uses only
Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024)
|
|
Ref |
Expression |
|
Hypotheses |
alcomw.1 |
⊢ ( 𝑥 = 𝑤 → ( 𝜑 ↔ 𝜓 ) ) |
|
|
alcomw.2 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜒 ) ) |
|
Assertion |
alcomw |
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 𝜑 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
alcomw.1 |
⊢ ( 𝑥 = 𝑤 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
alcomw.2 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
2
|
alcomiw |
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
4 |
1
|
alcomiw |
⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) |
5 |
3 4
|
impbii |
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 𝜑 ) |