Theorem nfccdeq 3784

Description: Variation of nfcdeq 3783 for classes. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2157. (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 2897	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eqid 2737	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3773	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3782	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3783	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2734	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  CondEqwcdeq 3769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-cleq 2729  df-clel 2816  df-nfc 2892  df-cdeq 3770

This theorem is referenced by: (None)