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Theorem elpreqpr 4364
Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
Assertion
Ref Expression
elpreqpr (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elpreqpr
StepHypRef Expression
1 elpri 4168 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 elex 3198 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
3 elpreqprlem 4363 . . . . 5 (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
4 eleq1 2686 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
5 preq1 4238 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝑥} = {𝐵, 𝑥})
65eqeq2d 2631 . . . . . . 7 (𝐴 = 𝐵 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝑥}))
76exbidv 1847 . . . . . 6 (𝐴 = 𝐵 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
84, 7imbi12d 334 . . . . 5 (𝐴 = 𝐵 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})))
93, 8mpbiri 248 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
109imp 445 . . 3 ((𝐴 = 𝐵𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
11 elpreqprlem 4363 . . . . . 6 (𝐶 ∈ V → ∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥})
12 prcom 4237 . . . . . . . 8 {𝐶, 𝐵} = {𝐵, 𝐶}
1312eqeq1i 2626 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥})
1413exbii 1771 . . . . . 6 (∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
1511, 14sylib 208 . . . . 5 (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
16 eleq1 2686 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
17 preq1 4238 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝑥} = {𝐶, 𝑥})
1817eqeq2d 2631 . . . . . . 7 (𝐴 = 𝐶 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥}))
1918exbidv 1847 . . . . . 6 (𝐴 = 𝐶 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥}))
2016, 19imbi12d 334 . . . . 5 (𝐴 = 𝐶 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})))
2115, 20mpbiri 248 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
2221imp 445 . . 3 ((𝐴 = 𝐶𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
2310, 22jaoian 823 . 2 (((𝐴 = 𝐵𝐴 = 𝐶) ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
241, 2, 23syl2anc 692 1 (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by:  elpreqprb  4365
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