Theorem rexrab 3692

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 3683	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	anbi1i 624	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒))
4		anass 469	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bitri 274	. 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
6	5	rexbii2 3090	1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃wrex 3070  {crab 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-rab 3433  df-v 3476

This theorem is referenced by:  wereu2  5673  frpomin  6341  wdom2d  9577  enfin2i  10318  infm3  12177  pmtrfrn  19367  pgpssslw  19523  ellspd  21576  1stcfb  23169  xkobval  23310  xkococn  23384  imasdsf1olem  24099  eqscut2  27532  scutun12  27536  cuteq0  27558  rusgrnumwwlks  29483  cvmliftlem15  34575  wsuclem  35089  poimirlem4  36795  poimirlem26  36817  poimirlem27  36818  infdesc  41687  rexrabdioph  41834  hbtlem6  42173