Theorem nfceqi 2748

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 1721	. . 3 ⊢ Ⅎ𝑥⊤
2		nfceqi.1	. . . 4 ⊢ 𝐴 = 𝐵
3	2	a1i 11	. . 3 ⊢ (⊤ → 𝐴 = 𝐵)
4	1, 3	nfceqdf 2747	. 2 ⊢ (⊤ → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	4	trud 1484	1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)

Syntax hints:   ↔ wb 195   = wceq 1475  ⊤wtru 1476  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740

This theorem is referenced by:  nfcxfr  2749  nfcxfrd  2750