Theorem nfbii 1569

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 1566	. . . 4 ⊢ (∀xφ ↔ ∀xψ)
3	1, 2	imbi12i 316	. . 3 ⊢ ((φ → ∀xφ) ↔ (ψ → ∀xψ))
4	3	albii 1566	. 2 ⊢ (∀x(φ → ∀xφ) ↔ ∀x(ψ → ∀xψ))
5		df-nf 1545	. 2 ⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
6		df-nf 1545	. 2 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
7	4, 5, 6	3bitr4i 268	1 ⊢ (Ⅎxφ ↔ Ⅎxψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-nf 1545

This theorem is referenced by:  nfxfr  1570  nfxfrd  1571  nfceqi  2486  dfnfc2  3910