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Theorem csbhypf 3171
 Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2904 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 = A[y / x]B = 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 2500 . . 3 x y = A
3 nfcsb1v 3168 . . . 4 x[y / x]B
4 csbhypf.2 . . . 4 xC
53, 4nfeq 2496 . . 3 x[y / x]B = C
62, 5nfim 1813 . 2 x(y = A[y / x]B = C)
7 eqeq1 2359 . . 3 (x = y → (x = Ay = A))
8 csbeq1a 3144 . . . 4 (x = yB = [y / x]B)
98eqeq1d 2361 . . 3 (x = y → (B = C[y / x]B = C))
107, 9imbi12d 311 . 2 (x = y → ((x = AB = C) ↔ (y = A[y / x]B = C)))
11 csbhypf.3 . 2 (x = AB = C)
126, 10, 11chvar 1986 1 (y = A[y / x]B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  Ⅎwnfc 2476  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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