Theorem sprid 48206

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 3482	. . 3 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2		rexv 3482	. . . 4 ⊢ (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
3	2	exbii 1878	. . 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 2830	1 ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:   = wceq 1570  ∃wex 1809  {cab 2741  ∃wrex 3089  Vcvv 3455  {cpr 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-v 3457

This theorem is referenced by: (None)