Theorem nfceqi 2304

Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

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 2233	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3	2	nfbii 1461	. . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)
4	3	albii 1458	. 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
5		df-nfc 2297	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
6		df-nfc 2297	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
7	4, 5, 6	3bitr4i 211	1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Syntax hints:   ↔ wb 104  ∀wal 1341   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfcxfr  2305  nfcxfrd  2306