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Theorem reu7 3631
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 3626 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)))
2 rmo4.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3 equequ1 2037 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
4 equcom 2030 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4bitrdi 290 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
62, 5imbi12d 348 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑧 = 𝑦)))
76cbvralvw 3349 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∀𝑦𝐴 (𝜓𝑧 = 𝑦))
87rexbii 3161 . . . 4 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦))
9 equequ1 2037 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
109imbi2d 344 . . . . . 6 (𝑧 = 𝑥 → ((𝜓𝑧 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1110ralbidv 3109 . . . . 5 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1211cbvrexvw 3350 . . . 4 (∃𝑧𝐴𝑦𝐴 (𝜓𝑧 = 𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
138, 12bitri 278 . . 3 (∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧) ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦))
1413anbi2i 626 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑧𝐴𝑥𝐴 (𝜑𝑥 = 𝑧)) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
151, 14bitri 278 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wral 3053  wrex 3054  ∃!wreu 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-10 2145  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clel 2811  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061
This theorem is referenced by:  cshwrepswhash1  16541
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