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Theorem sprid 45756
Description: Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprid {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Proof of Theorem sprid
StepHypRef Expression
1 rexv 3472 . . 3 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2 rexv 3472 . . . 4 (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
32exbii 1851 . . 3 (∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
41, 3bitri 275 . 2 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
54abbii 2803 1 {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  {cab 2710  wrex 3070  Vcvv 3447  {cpr 4592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-v 3449
This theorem is referenced by: (None)
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