Theorem drnf2

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 5-May-2018) Usage of this theorem is discouraged because it depends on ax-13 . Use nfbidv instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	drnf2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 ↔ Ⅎ 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	nfbidf	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 ↔ Ⅎ 𝑧 𝜓 ) )