Theorem nfbii 1859

Description: Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1791 changed. (Revised by Wolf Lammen, 12-Sep-2021.)

Hypothesis

Ref	Expression
nfbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
nfbii	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbii

Step	Hyp	Ref	Expression
1		nfbiit 1858	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
2		nfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	mpg 1804	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   ↔ wb 207  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

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfxfr  1860  nfxfrd  1861  dvelimhw  2353  nfeqf1  2387  nfceqi  2898  dfnfc2  4860  nfan1c  35255  bj-dvelimdv1  37205  bj-nfcf  37276  iunconnlem2  45378