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Theorem preqsnd 4499
Description: Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Hypotheses
Ref Expression
preqsnd.1 (𝜑𝐴 ∈ V)
preqsnd.2 (𝜑𝐵 ∈ V)
preqsnd.3 (𝜑𝐶 ∈ V)
Assertion
Ref Expression
preqsnd (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))

Proof of Theorem preqsnd
StepHypRef Expression
1 preqsnd.1 . 2 (𝜑𝐴 ∈ V)
2 preqsnd.2 . 2 (𝜑𝐵 ∈ V)
3 preqsnd.3 . 2 (𝜑𝐶 ∈ V)
4 dfsn2 4298 . . . 4 {𝐶} = {𝐶, 𝐶}
54eqeq2i 2736 . . 3 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
6 preq12bg 4493 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶))))
7 oridm 537 . . . 4 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
86, 7syl6bb 276 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
95, 8syl5bb 272 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
101, 2, 3, 3, 9syl22anc 1440 1 (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  {csn 4285  {cpr 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-un 3685  df-sn 4286  df-pr 4288
This theorem is referenced by:  1loopgrnb0  26529  disjdifprg  29616
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