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Theorem rexrab 3635
 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 → (𝜑𝜓))
Assertion
Ref Expression
rexrab (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
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
StepHypRef Expression
1 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
21elrab 3628 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32anbi1i 626 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ ((𝑥𝐴𝜓) ∧ 𝜒))
4 anass 472 . . 3 (((𝑥𝐴𝜓) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
53, 4bitri 278 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
65rexbii2 3208 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∃wrex 3107  {crab 3110 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-rab 3115  df-v 3443 This theorem is referenced by:  wereu2  5517  wdom2d  9031  enfin2i  9735  infm3  11590  pmtrfrn  18582  pgpssslw  18735  ellspd  20496  1stcfb  22060  xkobval  22201  xkococn  22275  imasdsf1olem  22990  rusgrnumwwlks  27770  cvmliftlem15  32673  frpomin  33206  wsuclem  33240  scutun12  33399  poimirlem4  35080  poimirlem26  35102  poimirlem27  35103  rexrabdioph  39778  hbtlem6  40116
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