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Theorem elunirab 3620
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 3619 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
2 df-rab 2332 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32unieqi 3617 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
43eleq2i 2120 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
5 df-rex 2329 . . 3 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)))
6 an12 503 . . . 4 ((𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ (𝐴𝑥 ∧ (𝑥𝐵𝜑)))
76exbii 1512 . . 3 (∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
85, 7bitri 177 . 2 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
91, 4, 83bitr4i 205 1 (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  wcel 1409  {cab 2042  wrex 2324  {crab 2327   cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-rab 2332  df-v 2576  df-uni 3608
This theorem is referenced by: (None)
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