Theorem nfbii 1466

Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
nfbii	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbii

Step	Hyp	Ref	Expression
1		nfbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	albii 1463	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
3	1, 2	imbi12i 238	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
4	3	albii 1463	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1454	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		df-nf 1454	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
7	4, 5, 6	3bitr4i 211	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  nfxfr  1467  nfxfrd  1468  nfsb  1939  nfsbt  1969  hbsbd  1975  sbal1yz  1994  dvelimALT  2003  dvelimfv  2004  dvelimor  2011  nfeudv  2034  nfeuv  2037  nfceqi  2308  nfreudxy  2643  dfnfc2  3814