Theorem nfceqi 2924

Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2216. (Revised by Wolf Lammen, 19-Jun-2023.)

Hypothesis

Ref	Expression
nfceqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
nfceqi	⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Proof of Theorem nfceqi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfceqi.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2857	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3	2	nfbii 1885	. . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)
4	3	albii 1852	. 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
5		df-nfc 2914	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
6		df-nfc 2914	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
7	4, 5, 6	3bitr4i 306	1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-cleq 2757  df-clel 2840  df-nfc 2914

This theorem is used by:  nfcxfr  2925  nfcxfrd  2926