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 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Proof of Theorem sprid

Step	Hyp	Ref	Expression
1		rexv 3433	. . 3 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2		rexv 3433	. . . 4 ⊢ (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
3	2	exbii 1855	. . 3 ⊢ (∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
4	1, 3	bitri 278	. 2 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
5	4	abbii 2808	1 ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

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)