Theorem ralab2 3002

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

Hypothesis

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

Assertion

Ref	Expression
ralab2	⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀y(φ → χ))

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

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

Proof of Theorem ralab2

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀x(x ∈ {y ∣ φ} → ψ))
2		nfsab1 2343	. . . 4 ⊢ Ⅎy x ∈ {y ∣ φ}
3		nfv 1619	. . . 4 ⊢ Ⅎyψ
4	2, 3	nfim 1813	. . 3 ⊢ Ⅎy(x ∈ {y ∣ φ} → ψ)
5		nfv 1619	. . 3 ⊢ Ⅎx(φ → χ)
6		eleq1 2413	. . . . 5 ⊢ (x = y → (x ∈ {y ∣ φ} ↔ y ∈ {y ∣ φ}))
7		abid 2341	. . . . 5 ⊢ (y ∈ {y ∣ φ} ↔ φ)
8	6, 7	syl6bb 252	. . . 4 ⊢ (x = y → (x ∈ {y ∣ φ} ↔ φ))
9		ralab2.1	. . . 4 ⊢ (x = y → (ψ ↔ χ))
10	8, 9	imbi12d 311	. . 3 ⊢ (x = y → ((x ∈ {y ∣ φ} → ψ) ↔ (φ → χ)))
11	4, 5, 10	cbval 1984	. 2 ⊢ (∀x(x ∈ {y ∣ φ} → ψ) ↔ ∀y(φ → χ))
12	1, 11	bitri 240	1 ⊢ (∀x ∈ {y ∣ φ}ψ ↔ ∀y(φ → χ))

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

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-clel 2349  df-ral 2620

This theorem is referenced by:  ralrab2  3003  ssintab  3944