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Theorem elpreqprlem 4871
Description: Lemma for elpreqpr 4872. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)
Assertion
Ref Expression
elpreqprlem (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elpreqprlem
StepHypRef Expression
1 eqid 2735 . . . 4 {𝐵, 𝐶} = {𝐵, 𝐶}
2 preq2 4739 . . . . . 6 (𝑥 = 𝐶 → {𝐵, 𝑥} = {𝐵, 𝐶})
32eqeq2d 2746 . . . . 5 (𝑥 = 𝐶 → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝐶}))
43spcegv 3597 . . . 4 (𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
51, 4mpi 20 . . 3 (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
65a1d 25 . 2 (𝐶 ∈ V → (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
7 dfsn2 4644 . . . 4 {𝐵} = {𝐵, 𝐵}
8 preq2 4739 . . . . . 6 (𝑥 = 𝐵 → {𝐵, 𝑥} = {𝐵, 𝐵})
98eqeq2d 2746 . . . . 5 (𝑥 = 𝐵 → ({𝐵} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝐵}))
109spcegv 3597 . . . 4 (𝐵𝑉 → ({𝐵} = {𝐵, 𝐵} → ∃𝑥{𝐵} = {𝐵, 𝑥}))
117, 10mpi 20 . . 3 (𝐵𝑉 → ∃𝑥{𝐵} = {𝐵, 𝑥})
12 prprc2 4771 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
1312eqeq1d 2737 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝑥}))
1413exbidv 1919 . . 3 𝐶 ∈ V → (∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥} ↔ ∃𝑥{𝐵} = {𝐵, 𝑥}))
1511, 14imbitrrid 246 . 2 𝐶 ∈ V → (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
166, 15pm2.61i 182 1 (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  elpreqpr  4872
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