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Theorem reu7 3399
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 3394 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)))
2 rmo4.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3 equequ1 1951 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
4 equcom 1944 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4syl6bb 276 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
62, 5imbi12d 334 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑧 = 𝑦)))
76cbvralv 3169 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∀𝑦𝐴 (𝜓𝑧 = 𝑦))
87rexbii 3039 . . . 4 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦))
9 equequ1 1951 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
109imbi2d 330 . . . . . 6 (𝑧 = 𝑥 → ((𝜓𝑧 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1110ralbidv 2985 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1211cbvrexv 3170 . . . 4 (∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
138, 12bitri 264 . . 3 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
1413anbi2i 730 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
151, 14bitri 264 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wral 2911  wrex 2912  ∃!wreu 2913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919
This theorem is referenced by:  cshwrepswhash1  15803
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