Theorem nfceqdf 2338

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 2266	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	nfbidf 1553	. . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵))
5	4	albidv 1838	. 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵))
6		df-nfc 2328	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
7		df-nfc 2328	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
8	5, 6, 7	3bitr4g 223	1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-nfc 2328

This theorem is referenced by:  nfopd  3825  dfnfc2  3857  nfimad  5018  nffvd  5570