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Theorem csbhypf 3856
 Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3500 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 2972 . . 3 𝑥 𝑦 = 𝐴
3 nfcsb1v 3852 . . . 4 𝑥𝑦 / 𝑥𝐵
4 csbhypf.2 . . . 4 𝑥𝐶
53, 4nfeq 2968 . . 3 𝑥𝑦 / 𝑥𝐵 = 𝐶
62, 5nfim 1897 . 2 𝑥(𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
7 eqeq1 2802 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
8 csbeq1a 3842 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
98eqeq1d 2800 . . 3 (𝑥 = 𝑦 → (𝐵 = 𝐶𝑦 / 𝑥𝐵 = 𝐶))
107, 9imbi12d 348 . 2 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)))
11 csbhypf.3 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
126, 10, 11chvarfv 2240 1 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2936  ⦋csb 3828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721  df-csb 3829 This theorem is referenced by:  disji2  5013  disjprgw  5026  disjprg  5027  disjxun  5029  tfisi  7556  coe1fzgsumdlem  20940  evl1gsumdlem  20990  iundisj2  24163  disji2f  30350  disjif2  30354  iundisj2f  30363  iundisj2fi  30556
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