Theorem rexdifsn 3844

Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Assertion

Ref	Expression
rexdifsn	⊢ (∃x ∈ (A ∖ {B})φ ↔ ∃x ∈ A (x ≠ B ∧ φ))

Proof of Theorem rexdifsn

Step	Hyp	Ref	Expression
1		eldifsn 3840	. . . 4 ⊢ (x ∈ (A ∖ {B}) ↔ (x ∈ A ∧ x ≠ B))
2	1	anbi1i 676	. . 3 ⊢ ((x ∈ (A ∖ {B}) ∧ φ) ↔ ((x ∈ A ∧ x ≠ B) ∧ φ))
3		anass 630	. . 3 ⊢ (((x ∈ A ∧ x ≠ B) ∧ φ) ↔ (x ∈ A ∧ (x ≠ B ∧ φ)))
4	2, 3	bitri 240	. 2 ⊢ ((x ∈ (A ∖ {B}) ∧ φ) ↔ (x ∈ A ∧ (x ≠ B ∧ φ)))
5	4	rexbii2 2644	1 ⊢ (∃x ∈ (A ∖ {B})φ ↔ ∃x ∈ A (x ≠ B ∧ φ))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∖ cdif 3207  {csn 3738

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-nan 1288  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-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-sn 3742

This theorem is referenced by: (None)