Theorem rexin 4098

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 4053	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	anbi1i 614	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑))
3		anass 461	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
4	2, 3	bitri 267	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
5	4	rexbii2 3186	1 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   ↔ wb 198   ∧ wa 387   ∈ wcel 2048  ∃wrex 3083   ∩ cin 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-v 3411  df-in 3832

This theorem is referenced by:  wefrc  5394  elidinxp  5750  bnd2  9108  subislly  21783  pcmplfin  30725  imaindm  32482