Theorem nfbii 1853

Description: Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1785 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 1852	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
2		nfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	mpg 1798	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   ↔ wb 205  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 206  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfxfr  1854  nfxfrd  1855  dvelimhw  2340  nfeqf1  2377  nfceqi  2899  nfra2wOLD  3296  dfnfc2  4933  bj-dvelimdv1  36047  bj-nfcf  36119  iunconnlem2  44011