Theorem wl-dfcleq 37848

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.) Base on wl-dfcleq.just 37844. (Revised by Wolf Lammen, 7-Apr-2026.)

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 𝐶 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		eleq1w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
3	1, 2	bibi12d 345	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)))
4	3	cbvalvw 2038	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5		eqid 2737	. 2 ⊢ 𝐴 = 𝐴
6		eqtr 2757	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
7	6	eqcomd 2743	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐶 = 𝐴)
8	7	ex 412	. 2 ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐶 = 𝐴))
9		eqeq2 2749	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
10	9	biimpd 229	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 → 𝑥 = 𝐵))
11	10	anim1d 612	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶)))
12	11	eximdv 1919	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶) → ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶)))
13		dfclel 2813	. . 3 ⊢ (𝐴 ∈ 𝐶 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶))
14		dfclel 2813	. . 3 ⊢ (𝐵 ∈ 𝐶 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶))
15	12, 13, 14	3imtr4g 296	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶))
16		wl-dfcleq.basic 37843	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
17		biimp 215	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
18	17	alimi 1813	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
19	1	biimpd 229	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
20	19	eqcoms 2745	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
21		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
22	21	biimpd 229	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 → 𝑥 ∈ 𝐵))
23	20, 22	imim12d 81	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
24	23	spimvw 1988	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
25	18, 24	syl 17	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
26	16, 25	sylbi 217	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
27	26	anim2d 613	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
28	27	eximdv 1919	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
29		dfclel 2813	. . 3 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴))
30		dfclel 2813	. . 3 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))
31	28, 29, 30	3imtr4g 296	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
32	4, 5, 8, 15, 31	wl-dfcleq.just 37844	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ 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-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812

This theorem is referenced by: (None)