Theorem sprid 47831

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 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Proof of Theorem sprid

Step	Hyp	Ref	Expression
1		rexv 3470	. . 3 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2		rexv 3470	. . . 4 ⊢ (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
3	2	exbii 1850	. . 3 ⊢ (∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
4	1, 3	bitri 275	. 2 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
5	4	abbii 2804	1 ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:   = wceq 1542  ∃wex 1781  {cab 2715  ∃wrex 3062  Vcvv 3442  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3444

This theorem is referenced by: (None)