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Theorem sprid 43513
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 3518 . . 3 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2 rexv 3518 . . . 4 (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
32exbii 1839 . . 3 (∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
41, 3bitri 276 . 2 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
54abbii 2883 1 {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wex 1771  {cab 2796  wrex 3136  Vcvv 3492  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rex 3141  df-v 3494
This theorem is referenced by: (None)
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