Theorem bj-dfcleq 32086

Description: Proof of class extensionality from the axiom of set extensionality (ax-ext 2590) and the definition of class equality (bj-df-cleq 32085). (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 32084	. 2 ⊢ (𝑢 = 𝑣 ↔ ∀𝑤(𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑣))
2	1	bj-df-cleq 32085	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603

This theorem is referenced by: (None)