Theorem bj-dfcleq 32894

Description: Proof of class extensionality from the axiom of set extensionality (ax-ext 2602) and the definition of class equality (bj-df-cleq 32893). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfcleq	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfcleq

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-cleqhyp 32892	. . 3 ⊢ (𝑢 = 𝑣 ↔ ∀𝑤(𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑣))
2	1	gen2 1723	. 2 ⊢ ∀𝑢∀𝑣(𝑢 = 𝑣 ↔ ∀𝑤(𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑣))
3	2	bj-df-cleq 32893	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by: (None)