Description: Weak version of alcom . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017) (Proof shortened by Wolf Lammen, 28-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | alcomiw.1 | ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | alcomiw | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alcomiw.1 | ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | 1 | cbvalvw | ⊢ ( ∀ 𝑦 𝜑 ↔ ∀ 𝑧 𝜓 ) |
| 3 | 2 | biimpi | ⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑧 𝜓 ) |
| 4 | 3 | alimi | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑧 𝜓 ) |
| 5 | ax-5 | ⊢ ( ∀ 𝑥 ∀ 𝑧 𝜓 → ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 𝜓 ) | |
| 6 | 1 | biimprd | ⊢ ( 𝑦 = 𝑧 → ( 𝜓 → 𝜑 ) ) |
| 7 | 6 | equcoms | ⊢ ( 𝑧 = 𝑦 → ( 𝜓 → 𝜑 ) ) |
| 8 | 7 | spimvw | ⊢ ( ∀ 𝑧 𝜓 → 𝜑 ) |
| 9 | 8 | 2alimi | ⊢ ( ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 𝜓 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| 10 | 4 5 9 | 3syl | ⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |