Theorem nfbii 1884

Description: Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1816 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 1883	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
2		nfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	mpg 1829	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  nfxfr  1885  nfxfrd  1886  dvelimhw  2377  nfeqf1  2411  nfceqi  2922  dfnfc2  4891  nfan1c  35756  bj-dvelimdv1  37891  bj-nfcf  37962  iunconnlem2  46056