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Theorem riotass 5419
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 3195 . . . 4 ((AB x A φ ∃!x B φ) → ∃!x A φ)
2 riotasbc 5407 . . . 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 892 . . . . 5 ((AB x A φ ∃!x B φ) → AB)
5 riotacl 5406 . . . . . 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 2923 . . . 4 ((AB x A φ ∃!x B φ) → (x A φ) B)
8 simp3 894 . . . 4 ((AB x A φ ∃!x B φ) → ∃!x B φ)
9 nfriota1 5399 . . . . 5 x(x A φ)
109nfsbc1 2758 . . . . 5 x[(x A φ) / x]φ
11 sbceq1a 2750 . . . . 5 (x = (x A φ) → (φ[(x A φ) / x]φ))
129, 10, 11riota2f 5413 . . . 4 (((x A φ) B ∃!x B φ) → ([(x A φ) / x]φ ↔ (x B φ) = (x A φ)))
137, 8, 12syl2anc 393 . . 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 2027 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 873   = wceq 1228   wcel 1374  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:  moriotass  5420
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