Theorem nfccdeq 3466

Description: Variation of nfcdeq 3465 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)

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	nfcri 2787	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		equid 1985	. . . . 5 ⊢ 𝑧 = 𝑧
4	3	cdeqth 3455	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝑧 = 𝑧)
5		nfccdeq.2	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)
6	4, 5	cdeqel 3464	. . 3 ⊢ CondEq(𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
7	2, 6	nfcdeq 3465	. 2 ⊢ (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	7	eqriv 2648	1 ⊢ 𝐴 = 𝐵

Syntax hints:   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  CondEqwcdeq 3451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782  df-cdeq 3452

This theorem is referenced by: (None)