Theorem nfccdeq 3743

Description: Variation of nfcdeq 3742 for classes. Usage of this theorem is discouraged because it depends on ax-13 2405. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2193. (Revised by GG, 19-May-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfccdeq.1	⊢ Ⅎ𝑥𝐴
nfccdeq.2	⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
nfccdeq	⊢ 𝐴 = 𝐵

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem nfccdeq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfccdeq.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2918	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eqid 2764	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3732	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3741	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3742	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2761	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1562   ∈ wcel 2144  Ⅎwnfc 2911  CondEqwcdeq 3728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-13 2405  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1802  df-nf 1806  df-sb 2093  df-cleq 2756  df-clel 2839  df-nfc 2913  df-cdeq 3729

This theorem is referenced by: (None)