Theorem nfbidf 1774

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypotheses

Ref	Expression
nfbidf.1	⊢ Ⅎxφ
nfbidf.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
nfbidf	⊢ (φ → (Ⅎxψ ↔ Ⅎxχ))

Proof of Theorem nfbidf

Step	Hyp	Ref	Expression
1		nfbidf.1	. . 3 ⊢ Ⅎxφ
2		nfbidf.2	. . . 4 ⊢ (φ → (ψ ↔ χ))
3	1, 2	albid 1772	. . . 4 ⊢ (φ → (∀xψ ↔ ∀xχ))
4	2, 3	imbi12d 311	. . 3 ⊢ (φ → ((ψ → ∀xψ) ↔ (χ → ∀xχ)))
5	1, 4	albid 1772	. 2 ⊢ (φ → (∀x(ψ → ∀xψ) ↔ ∀x(χ → ∀xχ)))
6		df-nf 1545	. 2 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
7		df-nf 1545	. 2 ⊢ (Ⅎxχ ↔ ∀x(χ → ∀xχ))
8	5, 6, 7	3bitr4g 279	1 ⊢ (φ → (Ⅎxψ ↔ Ⅎxχ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is used by:  nfsb4t  2080  dvelimdf  2082  nfcjust  2478  nfceqdf  2489