Theorem nfceqdf 2489

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfceqdf.1	⊢ Ⅎxφ
nfceqdf.2	⊢ (φ → A = B)

Assertion

Ref	Expression
nfceqdf	⊢ (φ → (ℲxA ↔ ℲxB))

Proof of Theorem nfceqdf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfceqdf.1	. . . 4 ⊢ Ⅎxφ
2		nfceqdf.2	. . . . 5 ⊢ (φ → A = B)
3	2	eleq2d 2420	. . . 4 ⊢ (φ → (y ∈ A ↔ y ∈ B))
4	1, 3	nfbidf 1774	. . 3 ⊢ (φ → (Ⅎx y ∈ A ↔ Ⅎx y ∈ B))
5	4	albidv 1625	. 2 ⊢ (φ → (∀yℲx y ∈ A ↔ ∀yℲx y ∈ B))
6		df-nfc 2479	. 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A)
7		df-nfc 2479	. 2 ⊢ (ℲxB ↔ ∀yℲx y ∈ B)
8	5, 6, 7	3bitr4g 279	1 ⊢ (φ → (ℲxA ↔ ℲxB))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This theorem was proved from 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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  dfnfc2  3910  nfopd  4606  nfimad  4955  nffvd  5336