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 → ( 𝜑 ↔ 𝜓 ) )
Assertion	dral2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 ) )

Proof

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