Theorem nfbii 1885

Description: Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1817 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 1884	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
2		nfbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	mpg 1830	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  nfxfr  1886  nfxfrd  1887  dvelimhw  2379  nfeqf1  2413  nfceqi  2924  dfnfc2  4896  nfan1c  35528  bj-dvelimdv1  37546  bj-nfcf  37617  iunconnlem2  45703