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Theorem elpreqpr 4547
 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 4342 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 elex 3352 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
3 elpreqprlem 4546 . . . . 5 (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
4 eleq1 2827 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
5 preq1 4412 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝑥} = {𝐵, 𝑥})
65eqeq2d 2770 . . . . . . 7 (𝐴 = 𝐵 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝑥}))
76exbidv 1999 . . . . . 6 (𝐴 = 𝐵 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
84, 7imbi12d 333 . . . . 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 444 . . 3 ((𝐴 = 𝐵𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
11 elpreqprlem 4546 . . . . . 6 (𝐶 ∈ V → ∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥})
12 prcom 4411 . . . . . . . 8 {𝐶, 𝐵} = {𝐵, 𝐶}
1312eqeq1i 2765 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥})
1413exbii 1923 . . . . . 6 (∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
1511, 14sylib 208 . . . . 5 (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
16 eleq1 2827 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
17 preq1 4412 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝑥} = {𝐶, 𝑥})
1817eqeq2d 2770 . . . . . . 7 (𝐴 = 𝐶 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥}))
1918exbidv 1999 . . . . . 6 (𝐴 = 𝐶 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥}))
2016, 19imbi12d 333 . . . . 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 444 . . 3 ((𝐴 = 𝐶𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
2310, 22jaoian 859 . 2 (((𝐴 = 𝐵𝐴 = 𝐶) ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
241, 2, 23syl2anc 696 1 (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340  {cpr 4323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324 This theorem is referenced by:  elpreqprb  4548
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