Theorem drnf2 2478

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 2406. Use nfbidv 1955 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
dral1.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
drnf2	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Proof of Theorem drnf2

Step	Hyp	Ref	Expression
1		nfae 2467	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		dral1.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	nfbidf 2263	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  nfsb4t  2533  drnfc2  2947