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Theorem reu7 2925
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
reu7 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu7
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 reu3 2920 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)))
2 rmo4.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3 equequ1 1705 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
4 equcom 1699 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4bitrdi 195 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
62, 5imbi12d 233 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑧 = 𝑦)))
76cbvralv 2696 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∀𝑦𝐴 (𝜓𝑧 = 𝑦))
87rexbii 2477 . . . 4 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦))
9 equequ1 1705 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
109imbi2d 229 . . . . . 6 (𝑧 = 𝑥 → ((𝜓𝑧 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1110ralbidv 2470 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1211cbvrexv 2697 . . . 4 (∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
138, 12bitri 183 . . 3 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
1413anbi2i 454 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
151, 14bitri 183 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wral 2448  wrex 2449  ∃!wreu 2450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456
This theorem is referenced by: (None)
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