Theorem rexab 3000

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,y   ψ,y

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

Proof of Theorem rexab

Step	Hyp	Ref	Expression
1		df-rex 2621	. 2 ⊢ (∃x ∈ {y ∣ φ}χ ↔ ∃x(x ∈ {y ∣ φ} ∧ χ))
2		vex 2863	. . . . 5 ⊢ x ∈ V
3		ralab.1	. . . . 5 ⊢ (y = x → (φ ↔ ψ))
4	2, 3	elab 2986	. . . 4 ⊢ (x ∈ {y ∣ φ} ↔ ψ)
5	4	anbi1i 676	. . 3 ⊢ ((x ∈ {y ∣ φ} ∧ χ) ↔ (ψ ∧ χ))
6	5	exbii 1582	. 2 ⊢ (∃x(x ∈ {y ∣ φ} ∧ χ) ↔ ∃x(ψ ∧ χ))
7	1, 6	bitri 240	1 ⊢ (∃x ∈ {y ∣ φ}χ ↔ ∃x(ψ ∧ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ 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-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-rex 2621  df-v 2862

This theorem is referenced by:  opeq  4620  frds  5936