Description: Obsolete version of dral1v as of 18-Nov-2024. (Contributed by NM, 24-Nov-1994) (Revised by BJ, 17-Jun-2019) Base the proof on ax12v . (Revised by Wolf Lammen, 30-Mar-2024) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | dral1v.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
Assertion | dral1vOLD | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dral1v.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
2 | nfa1 | ⊢ Ⅎ 𝑥 ∀ 𝑥 𝑥 = 𝑦 | |
3 | 2 1 | albid | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) ) |
4 | axc11v | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) ) | |
5 | axc11rv | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜓 ) ) | |
6 | 4 5 | impbid | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜓 ) ) |
7 | 3 6 | bitrd | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) ) |