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Theorem elunirab 3593
 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 3592 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
2 df-rab 2315 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32unieqi 3590 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
43eleq2i 2104 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
5 df-rex 2312 . . 3 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)))
6 an12 495 . . . 4 ((𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ (𝐴𝑥 ∧ (𝑥𝐵𝜑)))
76exbii 1496 . . 3 (∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
85, 7bitri 173 . 2 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
91, 4, 83bitr4i 201 1 (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2307  {crab 2310  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-uni 3581 This theorem is referenced by: (None)
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