Theorem nfceqdf 2895

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2116 and df-clel 2812. (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 2733	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5	1, 4	nfbidf 2232	. . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵))
6	5	albidv 1922	. 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵))
7		df-nfc 2886	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
8		df-nfc 2886	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-cleq 2729  df-nfc 2886

This theorem is referenced by:  nfopd  4847  dfnfc2  4886  nfimad  6029  nffvd  6847  riotasv2d  39254  nfcxfrdf  39263  nfded  39264  nfded2  39265  nfunidALT2  39266