Theorem rexrab 3370

Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab.1	⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexrab	⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rexrab

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
2	1	elrab 3363	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	anbi1i 731	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒))
4		anass 681	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bitri 264	. 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
6	5	rexbii2 3039	1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  ∃wrex 2913  {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202

This theorem is referenced by:  wereu2  5111  wdom2d  8485  enfin2i  9143  infm3  10982  pmtrfrn  17878  pgpssslw  18029  ellspd  20141  1stcfb  21248  xkobval  21389  xkococn  21463  imasdsf1olem  22178  rusgrnumwwlks  26869  cvmliftlem15  31280  wsuclem  31773  wsuclemOLD  31774  scutun12  31917  poimirlem4  33413  poimirlem26  33435  poimirlem27  33436  rexrabdioph  37358  hbtlem6  37699