Theorem nfbii 1473

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 1470	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
3	1, 2	imbi12i 239	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
4	3	albii 1470	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1461	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		df-nf 1461	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
7	4, 5, 6	3bitr4i 212	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351  Ⅎwnf 1460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  nfxfr  1474  nfxfrd  1475  nfsb  1946  nfsbt  1976  hbsbd  1982  sbal1yz  2001  dvelimALT  2010  dvelimfv  2011  dvelimor  2018  nfeudv  2041  nfeuv  2044  nfceqi  2315  nfreudxy  2651  dfnfc2  3829