Theorem nfceqdf 2904

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2112 and df-clel 2818. (Revised by WL and SN, 23-Aug-2024.)

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		eleq2w2 2736	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5	1, 4	nfbidf 2221	. . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵))
6	5	albidv 1927	. 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵))
7		df-nfc 2891	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
8		df-nfc 2891	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2110  Ⅎwnfc 2889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-nf 1791  df-cleq 2732  df-nfc 2891

This theorem is referenced by:  nfopd  4827  dfnfc2  4869  nfimad  5976  nffvd  6781  riotasv2d  36965  nfcxfrdf  36974  nfded  36975  nfded2  36976  nfunidALT2  36977