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Theorem sprid 44599
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 3433 . . 3 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2 rexv 3433 . . . 4 (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
32exbii 1855 . . 3 (∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
41, 3bitri 278 . 2 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
54abbii 2808 1 {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wex 1787  {cab 2714  wrex 3062  Vcvv 3408  {cpr 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3067  df-v 3410
This theorem is referenced by: (None)
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