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Theorem moriotass 7420
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
moriotass ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem moriotass
StepHypRef Expression
1 ssrexv 4065 . . . . 5 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
21imp 406 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑) → ∃𝑥𝐵 𝜑)
323adant3 1131 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → ∃𝑥𝐵 𝜑)
4 simp3 1137 . . 3 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → ∃*𝑥𝐵 𝜑)
5 reu5 3380 . . 3 (∃!𝑥𝐵 𝜑 ↔ (∃𝑥𝐵 𝜑 ∧ ∃*𝑥𝐵 𝜑))
63, 4, 5sylanbrc 583 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → ∃!𝑥𝐵 𝜑)
7 riotass 7419 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
86, 7syld3an3 1408 1 ((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  wss 3963  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by: (None)
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