ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqsbc3r Structured version   GIF version

Theorem eqsbc3r 2796
Description: eqsbc3 2779 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r (A B → ([A / x]𝐶 = x𝐶 = A))
Distinct variable groups:   x,𝐶   x,A
Allowed substitution hint:   B(x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2024 . . . . . 6 (𝐶 = xx = 𝐶)
21sbcbii 2795 . . . . 5 ([A / x]𝐶 = x[A / x]x = 𝐶)
32biimpi 113 . . . 4 ([A / x]𝐶 = x[A / x]x = 𝐶)
4 eqsbc3 2779 . . . 4 (A B → ([A / x]x = 𝐶A = 𝐶))
53, 4syl5ib 143 . . 3 (A B → ([A / x]𝐶 = xA = 𝐶))
6 eqcom 2024 . . 3 (A = 𝐶𝐶 = A)
75, 6syl6ib 150 . 2 (A B → ([A / x]𝐶 = x𝐶 = A))
8 idd 21 . . . . 5 (A B → (𝐶 = A𝐶 = A))
98, 6syl6ibr 151 . . . 4 (A B → (𝐶 = AA = 𝐶))
109, 4sylibrd 158 . . 3 (A B → (𝐶 = A[A / x]x = 𝐶))
1110, 2syl6ibr 151 . 2 (A B → (𝐶 = A[A / x]𝐶 = x))
127, 11impbid 120 1 (A B → ([A / x]𝐶 = x𝐶 = A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  [wsbc 2741
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-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator