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Theorem rmo4 2850
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
rmo4 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rmo4
StepHypRef Expression
1 df-rmo 2401 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 an4 560 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)))
3 ancom 264 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
43anbi1i 453 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
52, 4bitri 183 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
65imbi1i 237 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦))
7 impexp 261 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)))
8 impexp 261 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
96, 7, 83bitri 205 . . . . . 6 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
109albii 1431 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
11 df-ral 2398 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
12 r19.21v 2486 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1310, 11, 123bitr2i 207 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1413albii 1431 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
15 eleq1 2180 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 rmo4.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16anbi12d 464 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817mo4 2038 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦))
19 df-ral 2398 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
2014, 18, 193bitr4i 211 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
211, 20bitri 183 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314  wcel 1465  ∃*wmo 1978  wral 2393  ∃*wrmo 2396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-cleq 2110  df-clel 2113  df-ral 2398  df-rmo 2401
This theorem is referenced by:  reu4  2851  disjnim  3890  supmoti  6848  lteupri  7393  elrealeu  7605  rereceu  7665  exbtwnz  9996  rsqrmo  10767  divalglemeunn  11545  divalglemeuneg  11547  bezoutlemeu  11622  pw2dvdseu  11773  dedekindeu  12697  dedekindicclemicc  12706
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