ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotass Structured version   GIF version

Theorem riotass 5438
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotass ((AB x A φ ∃!x B φ) → (x A φ) = (x B φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem riotass
StepHypRef Expression
1 reuss 3212 . . . 4 ((AB x A φ ∃!x B φ) → ∃!x A φ)
2 riotasbc 5426 . . . 4 (∃!x A φ[(x A φ) / x]φ)
31, 2syl 14 . . 3 ((AB x A φ ∃!x B φ) → [(x A φ) / x]φ)
4 simp1 903 . . . . 5 ((AB x A φ ∃!x B φ) → AB)
5 riotacl 5425 . . . . . 6 (∃!x A φ → (x A φ) A)
61, 5syl 14 . . . . 5 ((AB x A φ ∃!x B φ) → (x A φ) A)
74, 6sseldd 2940 . . . 4 ((AB x A φ ∃!x B φ) → (x A φ) B)
8 simp3 905 . . . 4 ((AB x A φ ∃!x B φ) → ∃!x B φ)
9 nfriota1 5418 . . . . 5 x(x A φ)
109nfsbc1 2775 . . . . 5 x[(x A φ) / x]φ
11 sbceq1a 2767 . . . . 5 (x = (x A φ) → (φ[(x A φ) / x]φ))
129, 10, 11riota2f 5432 . . . 4 (((x A φ) B ∃!x B φ) → ([(x A φ) / x]φ ↔ (x B φ) = (x A φ)))
137, 8, 12syl2anc 391 . . 3 ((AB x A φ ∃!x B φ) → ([(x A φ) / x]φ ↔ (x B φ) = (x A φ)))
143, 13mpbid 135 . 2 ((AB x A φ ∃!x B φ) → (x B φ) = (x A φ))
1514eqcomd 2042 1 ((AB x A φ ∃!x B φ) → (x A φ) = (x B φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884   = wceq 1242   wcel 1390  wrex 2301  ∃!wreu 2302  [wsbc 2758  wss 2911  crio 5410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  moriotass  5439
  Copyright terms: Public domain W3C validator