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Theorem funsneqopsnOLD 6645
Description: Obsolete as of 15-Jul-2022. Use snopeqopsnid 5165 instead. (Contributed by AV, 24-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
funsndifnop.a 𝐴 ∈ V
funsndifnop.b 𝐵 ∈ V
funsndifnop.g 𝐺 = {⟨𝐴, 𝐵⟩}
Assertion
Ref Expression
funsneqopsnOLD (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)

Proof of Theorem funsneqopsnOLD
StepHypRef Expression
1 opeq2 4594 . . . 4 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
21sneqd 4380 . . 3 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3 funsndifnop.g . . 3 𝐺 = {⟨𝐴, 𝐵⟩}
42, 3syl6reqr 2852 . 2 (𝐴 = 𝐵𝐺 = {⟨𝐴, 𝐴⟩})
5 eqid 2799 . . . 4 𝐴 = 𝐴
6 eqid 2799 . . . 4 {𝐴} = {𝐴}
75, 6, 63pm3.2i 1439 . . 3 (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})
8 eqeq1 2803 . . . 4 (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩))
9 funsndifnop.a . . . . 5 𝐴 ∈ V
10 snex 5099 . . . . 5 {𝐴} ∈ V
119, 9, 10, 10snopeqopOLD 5162 . . . 4 ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
128, 11syl6bb 279 . . 3 (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})))
137, 12mpbiri 250 . 2 (𝐺 = {⟨𝐴, 𝐴⟩} → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
144, 13syl 17 1 (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  {csn 4368  cop 4374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375
This theorem is referenced by:  funsneqopOLD  6646
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