Theorem wl-dfcleq 37790

Description: The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2709) and the definition of class equality (df-cleq 2729). Its forward implication is called "class extensionality". Remark: the proof uses axextb 2712 to prove also the hypothesis of df-cleq 2729 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1797, equid 2014 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem wl-dfcleq

Dummy variables 𝑦 𝑧 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axextb 2712	. 2 ⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))
2		axextb 2712	. 2 ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))
3	1, 2	wl-df-cleq 37789	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729

This theorem is referenced by: (None)