Theorem rexab2 3004

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

Hypothesis

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

Assertion

Ref	Expression
rexab2	⊢ (∃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 rexab2

Step	Hyp	Ref	Expression
1		df-rex 2621	. 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	nfan 1824	. . 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	anbi12d 691	. . 3 ⊢ (x = y → ((x ∈ {y ∣ φ} ∧ ψ) ↔ (φ ∧ χ)))
11	4, 5, 10	cbvex 1985	. 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   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616

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-rex 2621

This theorem is referenced by:  rexrab2  3005