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Theorem csbhypf 2879
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2597 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 2186 . . 3 x y = A
3 nfcsb1v 2876 . . . 4 xy / xB
4 csbhypf.2 . . . 4 x𝐶
53, 4nfeq 2182 . . 3 xy / xB = 𝐶
62, 5nfim 1461 . 2 x(y = Ay / xB = 𝐶)
7 eqeq1 2043 . . 3 (x = y → (x = Ay = A))
8 csbeq1a 2854 . . . 4 (x = yB = y / xB)
98eqeq1d 2045 . . 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 1637 1 (y = Ay / xB = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wnfc 2162  csb 2846
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-sbc 2759  df-csb 2847
This theorem is referenced by:  tfisi  4253
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