Theorem ralrab2 3003

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	⊢ (x = y → (ψ ↔ χ))

Assertion

Ref	Expression
ralrab2	⊢ (∀x ∈ {y ∈ A ∣ φ}ψ ↔ ∀y ∈ A (φ → χ))

Distinct variable groups:   x,y   x,A   χ,x   φ,x   ψ,y

Allowed substitution hints:   φ(y)   ψ(x)   χ(y)   A(y)

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 ⊢ {y ∈ A ∣ φ} = {y ∣ (y ∈ A ∧ φ)}
2	1	raleqi 2812	. 2 ⊢ (∀x ∈ {y ∈ A ∣ φ}ψ ↔ ∀x ∈ {y ∣ (y ∈ A ∧ φ)}ψ)
3		ralab2.1	. . 3 ⊢ (x = y → (ψ ↔ χ))
4	3	ralab2 3002	. 2 ⊢ (∀x ∈ {y ∣ (y ∈ A ∧ φ)}ψ ↔ ∀y((y ∈ A ∧ φ) → χ))
5		impexp 433	. . . 4 ⊢ (((y ∈ A ∧ φ) → χ) ↔ (y ∈ A → (φ → χ)))
6	5	albii 1566	. . 3 ⊢ (∀y((y ∈ A ∧ φ) → χ) ↔ ∀y(y ∈ A → (φ → χ)))
7		df-ral 2620	. . 3 ⊢ (∀y ∈ A (φ → χ) ↔ ∀y(y ∈ A → (φ → χ)))
8	6, 7	bitr4i 243	. 2 ⊢ (∀y((y ∈ A ∧ φ) → χ) ↔ ∀y ∈ A (φ → χ))
9	2, 4, 8	3bitri 262	1 ⊢ (∀x ∈ {y ∈ A ∣ φ}ψ ↔ ∀y ∈ A (φ → χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624

This theorem is referenced by: (None)