Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotass2 Structured version   GIF version

Theorem riotass2 5418
 Description: Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
Assertion
Ref Expression
riotass2 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → (x A φ) = (x B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem riotass2
StepHypRef Expression
1 reuss2 3194 . . . 4 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x A φ)
2 simplr 470 . . . 4 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → x A (φψ))
3 riotasbc 5407 . . . . 5 (∃!x A φ[(x A φ) / x]φ)
4 riotacl 5406 . . . . . 6 (∃!x A φ → (x A φ) A)
5 rspsbc 2817 . . . . . . 7 ((x A φ) A → (x A (φψ) → [(x A φ) / x](φψ)))
6 sbcimg 2781 . . . . . . 7 ((x A φ) A → ([(x A φ) / x](φψ) ↔ ([(x A φ) / x]φ[(x A φ) / x]ψ)))
75, 6sylibd 138 . . . . . 6 ((x A φ) A → (x A (φψ) → ([(x A φ) / x]φ[(x A φ) / x]ψ)))
84, 7syl 14 . . . . 5 (∃!x A φ → (x A (φψ) → ([(x A φ) / x]φ[(x A φ) / x]ψ)))
93, 8mpid 37 . . . 4 (∃!x A φ → (x A (φψ) → [(x A φ) / x]ψ))
101, 2, 9sylc 56 . . 3 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → [(x A φ) / x]ψ)
111, 4syl 14 . . . . 5 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → (x A φ) A)
12 ssel 2916 . . . . . 6 (AB → ((x A φ) A → (x A φ) B))
1312ad2antrr 460 . . . . 5 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → ((x A φ) A → (x A φ) B))
1411, 13mpd 13 . . . 4 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → (x A φ) B)
15 simprr 472 . . . 4 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x B ψ)
16 nfriota1 5399 . . . . 5 x(x A φ)
1716nfsbc1 2758 . . . . 5 x[(x A φ) / x]ψ
18 sbceq1a 2750 . . . . 5 (x = (x A φ) → (ψ[(x A φ) / x]ψ))
1916, 17, 18riota2f 5413 . . . 4 (((x A φ) B ∃!x B ψ) → ([(x A φ) / x]ψ ↔ (x B ψ) = (x A φ)))
2014, 15, 19syl2anc 393 . . 3 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → ([(x A φ) / x]ψ ↔ (x B ψ) = (x A φ)))
2110, 20mpbid 135 . 2 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → (x B ψ) = (x A φ))
2221eqcomd 2027 1 (((AB x A (φψ)) (x A φ ∃!x B ψ)) → (x A φ) = (x B ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285  ∃!wreu 2286  [wsbc 2741   ⊆ wss 2894  ℩crio 5392 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator