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Theorem sprid 47399
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 3507 . . 3 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2 rexv 3507 . . . 4 (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
32exbii 1845 . . 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 2807 1 {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  {cab 2712  wrex 3068  Vcvv 3478  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480
This theorem is referenced by: (None)
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