Theorem nfccdeq 3750

Description: Variation of nfcdeq 3749 for classes. Usage of this theorem is discouraged because it depends on ax-13 2410. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2198. (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 2923	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eqid 2769	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3739	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3748	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3749	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2766	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1567   ∈ wcel 2149  Ⅎwnfc 2916  CondEqwcdeq 3735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-13 2410  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-cleq 2761  df-clel 2844  df-nfc 2918  df-cdeq 3736

This theorem is referenced by: (None)