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Theorem rabbi 2789

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2850. (Contributed by NM, 25-Nov-2013.)

**Assertion**
Ref	Expression
rabbi	⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

**Proof of Theorem rabbi**
Step	Hyp	Ref	Expression
1		abbi 2463	. 2 ⊢ (∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)) ↔ {x ∣ (x ∈ A ∧ ψ)} = {x ∣ (x ∈ A ∧ χ)})
2		df-ral 2619	. . 3 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ ∀x(x ∈ A → (ψ ↔ χ)))
3		pm5.32 617	. . . 4 ⊢ ((x ∈ A → (ψ ↔ χ)) ↔ ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
4	3	albii 1566	. . 3 ⊢ (∀x(x ∈ A → (ψ ↔ χ)) ↔ ∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
5	2, 4	bitri 240	. 2 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ ∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
6		df-rab 2623	. . 3 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)}
7		df-rab 2623	. . 3 ⊢ {x ∈ A ∣ χ} = {x ∣ (x ∈ A ∧ χ)}
8	6, 7	eqeq12i 2366	. 2 ⊢ ({x ∈ A ∣ ψ} = {x ∈ A ∣ χ} ↔ {x ∣ (x ∈ A ∧ ψ)} = {x ∣ (x ∈ A ∧ χ)})
9	1, 5, 8	3bitr4i 268	1 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 ∀wral 2614 {crab 2618

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-ral 2619 df-rab 2623

This theorem is referenced by: rabbidva 2850 fnpm 6008

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