Theorem nfccdeq 3678

Description: Variation of nfcdeq 3677 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2093. (Revised by Gino Giotto, 19-May-2023.)

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 2922	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eqid 2778	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3667	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3676	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3677	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2775	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1507   ∈ wcel 2050  Ⅎwnfc 2916  CondEqwcdeq 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016  df-cleq 2771  df-clel 2846  df-nfc 2918  df-cdeq 3664

This theorem is referenced by: (None)