Metamath Proof Explorer


Theorem dral1

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of Megill p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker dral1v if possible. (Contributed by NM, 24-Nov-1994) Remove dependency on ax-11 . (Revised by Wolf Lammen, 6-Sep-2018) (New usage is discouraged.)

Ref Expression
Hypothesis dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion dral1 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
2 nfa1 𝑥𝑥 𝑥 = 𝑦
3 2 1 albid ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) )
4 axc11 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) )
5 axc11r ( ∀ 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 ) )