Theorem rabbi 2790

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2851. (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		abbib 2464	. 2 ⊢ ({x ∣ (x ∈ A ∧ ψ)} = {x ∣ (x ∈ A ∧ χ)} ↔ ∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
2		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)}
3		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ χ} = {x ∣ (x ∈ A ∧ χ)}
4	2, 3	eqeq12i 2366	. 2 ⊢ ({x ∈ A ∣ ψ} = {x ∈ A ∣ χ} ↔ {x ∣ (x ∈ A ∧ ψ)} = {x ∣ (x ∈ A ∧ χ)})
5		df-ral 2620	. . 3 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ ∀x(x ∈ A → (ψ ↔ χ)))
6		pm5.32 617	. . . 4 ⊢ ((x ∈ A → (ψ ↔ χ)) ↔ ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
7	6	albii 1566	. . 3 ⊢ (∀x(x ∈ A → (ψ ↔ χ)) ↔ ∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
8	5, 7	bitri 240	. 2 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ ∀x((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
9	1, 4, 8	3bitr4ri 269	1 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  {crab 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2620  df-rab 2624

This theorem is referenced by:  rabbidva  2851  fnpm  6009