Theorem nfceqi 2236

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 2166	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3	2	nfbii 1417	. . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)
4	3	albii 1414	. 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
5		df-nfc 2229	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
6		df-nfc 2229	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
7	4, 5, 6	3bitr4i 211	1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Syntax hints:   ↔ wb 104  ∀wal 1297   = wceq 1299  Ⅎwnf 1404   ∈ wcel 1448  Ⅎwnfc 2227

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-17 1474  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-cleq 2093  df-clel 2096  df-nfc 2229

This theorem is referenced by:  nfcxfr  2237  nfcxfrd  2238