Theorem rexin 4239

Description: Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018.)

Assertion

Ref	Expression
rexin	⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))

Proof of Theorem rexin

Step	Hyp	Ref	Expression
1		elin 3964	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	anbi1i 623	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑))
3		anass 468	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
4	2, 3	bitri 275	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
5	4	rexbii2 3089	1 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2105  ∃wrex 3069   ∩ cin 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-in 3955

This theorem is referenced by:  wefrc  5670  elidinxp  6043  imaindm  6298  bnd2  9894  subislly  23305  pcmplfin  33304