Theorem drnf2 2443

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. Usage of nfbidv 1928 is preferred, which requires fewer axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.) (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 2432	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		dral1.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	nfbidf 2225	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-10 2144  ax-11 2161  ax-12 2178  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfsb4t  2503  drnfc2  2920  drnfc2OLD  2921