Theorem rexin 4150

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 4096	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	anbi1i 636	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑))
3		anass 474	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
4	2, 3	bitri 278	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
5	4	rexbii2 3109	1 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∃wrex 3090   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-v 3456  df-in 4094

This theorem is used by:  wefrc  5652  elidinxp  6054  imaindm  6328  bnd2  9993  subislly  23855  pcmplfin  34603  sswfaxreg  46241