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Theorem elunirab 3640
Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elunirab
StepHypRef Expression
1 eluniab 3639 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
2 df-rab 2362 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32unieqi 3637 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
43eleq2i 2149 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
5 df-rex 2359 . . 3 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)))
6 an12 526 . . . 4 ((𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ (𝐴𝑥 ∧ (𝑥𝐵𝜑)))
76exbii 1537 . . 3 (∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
85, 7bitri 182 . 2 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
91, 4, 83bitr4i 210 1 (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1422  wcel 1434  {cab 2069  wrex 2354  {crab 2357   cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-rab 2362  df-v 2614  df-uni 3628
This theorem is referenced by: (None)
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