Theorem nfbii 1948

Description: Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1880 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 1947	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
2		nfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	mpg 1893	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   ↔ wb 198  Ⅎwnf 1879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905

This theorem depends on definitions:  df-bi 199  df-ex 1876  df-nf 1880

This theorem is referenced by:  nfxfr  1949  nfxfrd  1950  dvelimhw  2360  nfeqf1  2387  dfnfc2  4649  bj-dvelimdv1  33328  bj-nfcf  33412  iunconnlem2  39926