Theorem nfbii 1461

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 1458	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
3	1, 2	imbi12i 238	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
4	3	albii 1458	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1449	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		df-nf 1449	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
7	4, 5, 6	3bitr4i 211	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448

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 1435  ax-gen 1437

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  nfxfr  1462  nfxfrd  1463  nfsb  1934  nfsbt  1964  hbsbd  1970  sbal1yz  1989  dvelimALT  1998  dvelimfv  1999  dvelimor  2006  nfeudv  2029  nfeuv  2032  nfceqi  2304  nfreudxy  2639  dfnfc2  3807