Theorem nfceqi 2485

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 2417	. . . 4 ⊢ (y ∈ A ↔ y ∈ B)
3	2	nfbii 1569	. . 3 ⊢ (Ⅎx y ∈ A ↔ Ⅎx y ∈ B)
4	3	albii 1566	. 2 ⊢ (∀yℲx y ∈ A ↔ ∀yℲx y ∈ B)
5		df-nfc 2478	. 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A)
6		df-nfc 2478	. 2 ⊢ (ℲxB ↔ ∀yℲx y ∈ B)
7	4, 5, 6	3bitr4i 268	1 ⊢ (ℲxA ↔ ℲxB)

Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfcxfr  2486  nfcxfrd  2487