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Theorem snopeqopOLD 5129
 Description: Obsolete version of snopeqop 5128 as of 15-Jul-2022 . (Contributed by AV, 18-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
snopeqop.a 𝐴 ∈ V
snopeqop.b 𝐵 ∈ V
propeqop.c 𝐶 ∈ V
propeqop.d 𝐷 ∈ V
Assertion
Ref Expression
snopeqopOLD ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))

Proof of Theorem snopeqopOLD
StepHypRef Expression
1 snopeqop.a . . . 4 𝐴 ∈ V
2 snopeqop.b . . . 4 𝐵 ∈ V
3 propeqop.c . . . 4 𝐶 ∈ V
41, 2, 3opeqsnOLD 5126 . . 3 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
54anbi2i 616 . 2 ((𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})))
6 eqcom 2772 . . 3 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩})
7 propeqop.d . . . 4 𝐷 ∈ V
8 opex 5090 . . . 4 𝐴, 𝐵⟩ ∈ V
93, 7, 8opeqsnOLD 5126 . . 3 (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
106, 9bitri 266 . 2 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
11 3anan12 1117 . 2 ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})))
125, 10, 113bitr4i 294 1 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  Vcvv 3350  {csn 4336  ⟨cop 4342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343 This theorem is referenced by:  funsneqopsnOLD  6613
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