Theorem rexdifsn 4762

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 4753	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	anbi1i 635	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) ∧ 𝜑))
3		anass 473	. . 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 3108	1 ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∖ cdif 3902  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rex 3090  df-v 3457  df-dif 3908  df-sn 4590

This theorem is referenced by:  symgfix2  19481  usgr2pth0  30114  wspniunwspnon  30272  dihatexv  42112  lcfl8b  42278