Theorem nfceqi 2761

Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Hypothesis

Ref	Expression
nfceqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
nfceqi	⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Proof of Theorem nfceqi

Step	Hyp	Ref	Expression
1		nftru 1730	. . 3 ⊢ Ⅎ𝑥⊤
2		nfceqi.1	. . . 4 ⊢ 𝐴 = 𝐵
3	2	a1i 11	. . 3 ⊢ (⊤ → 𝐴 = 𝐵)
4	1, 3	nfceqdf 2760	. 2 ⊢ (⊤ → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	4	trud 1493	1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Syntax hints:   ↔ wb 196   = wceq 1483  ⊤wtru 1484  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfcxfr  2762  nfcxfrd  2763