Description: Alternate proof of dral1 , shorter but requiring ax-11 . (Contributed by NM, 24-Nov-1994) (Proof shortened by Wolf Lammen, 22-Apr-2018) (New usage is discouraged.) (Proof modification is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | dral1.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
Assertion | dral1ALT | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dral1.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
2 | 1 | dral2 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) ) |
3 | axc11 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) ) | |
4 | axc11r | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜓 ) ) | |
5 | 3 4 | impbid | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜓 ) ) |
6 | 2 5 | bitrd | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) ) |