Theorem nfccdeq 3771

Description: Variation of nfcdeq 3770 for classes. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2161. (Revised by Gino Giotto, 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 groups:   𝑥,𝐵   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   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	nfcriv 2969	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eqid 2823	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3760	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3769	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3770	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2820	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2963  CondEqwcdeq 3756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-cleq 2816  df-clel 2895  df-nfc 2965  df-cdeq 3757

This theorem is referenced by: (None)