Metamath Proof Explorer


Theorem dral2

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 . Usage of albidv is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005) Allow a shortening of dral1 . (Revised by Wolf Lammen, 4-Mar-2018) (New usage is discouraged.)

Ref Expression
Hypothesis dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion dral2 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 ) )

Proof

Step Hyp Ref Expression
1 dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
2 nfae 𝑧𝑥 𝑥 = 𝑦
3 2 1 albid ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 ) )