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Theorem csbhypf 2862
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2580 for class substitution version. (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
csbhypf.1 xA
csbhypf.2 x𝐶
csbhypf.3 (x = AB = 𝐶)
Assertion
Ref Expression
csbhypf (y = Ay / xB = 𝐶)
Distinct variable group:   x,y
Allowed substitution hints:   A(x,y)   B(x,y)   𝐶(x,y)

Proof of Theorem csbhypf
StepHypRef Expression
1 csbhypf.1 . . . 4 xA
21nfeq2 2171 . . 3 x y = A
3 nfcsb1v 2859 . . . 4 xy / xB
4 csbhypf.2 . . . 4 x𝐶
53, 4nfeq 2167 . . 3 xy / xB = 𝐶
62, 5nfim 1446 . 2 x(y = Ay / xB = 𝐶)
7 eqeq1 2028 . . 3 (x = y → (x = Ay = A))
8 csbeq1a 2837 . . . 4 (x = yB = y / xB)
98eqeq1d 2030 . . 3 (x = y → (B = 𝐶y / xB = 𝐶))
107, 9imbi12d 223 . 2 (x = y → ((x = AB = 𝐶) ↔ (y = Ay / xB = 𝐶)))
11 csbhypf.3 . 2 (x = AB = 𝐶)
126, 10, 11chvar 1622 1 (y = Ay / xB = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wnfc 2147  csb 2829
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-sbc 2742  df-csb 2830
This theorem is referenced by:  tfisi  4237
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