Theorem rexrab 3000

Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab.1	⊢ (y = x → (φ ↔ ψ))

Assertion

Ref	Expression
rexrab	⊢ (∃x ∈ {y ∈ A ∣ φ}χ ↔ ∃x ∈ A (ψ ∧ χ))

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

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

Proof of Theorem rexrab

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 ⊢ (y = x → (φ ↔ ψ))
2	1	elrab 2994	. . . 4 ⊢ (x ∈ {y ∈ A ∣ φ} ↔ (x ∈ A ∧ ψ))
3	2	anbi1i 676	. . 3 ⊢ ((x ∈ {y ∈ A ∣ φ} ∧ χ) ↔ ((x ∈ A ∧ ψ) ∧ χ))
4		anass 630	. . 3 ⊢ (((x ∈ A ∧ ψ) ∧ χ) ↔ (x ∈ A ∧ (ψ ∧ χ)))
5	3, 4	bitri 240	. 2 ⊢ ((x ∈ {y ∈ A ∣ φ} ∧ χ) ↔ (x ∈ A ∧ (ψ ∧ χ)))
6	5	rexbii2 2643	1 ⊢ (∃x ∈ {y ∈ A ∣ φ}χ ↔ ∃x ∈ A (ψ ∧ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃wrex 2615  {crab 2618

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 2478  df-rex 2620  df-rab 2623  df-v 2861

This theorem is referenced by:  phialllem1  4616