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Theorem csbhypf 3745
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3439 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 2955 . . 3 𝑥 𝑦 = 𝐴
3 nfcsb1v 3742 . . . 4 𝑥𝑦 / 𝑥𝐵
4 csbhypf.2 . . . 4 𝑥𝐶
53, 4nfeq 2951 . . 3 𝑥𝑦 / 𝑥𝐵 = 𝐶
62, 5nfim 1996 . 2 𝑥(𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
7 eqeq1 2801 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
8 csbeq1a 3735 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
98eqeq1d 2799 . . 3 (𝑥 = 𝑦 → (𝐵 = 𝐶𝑦 / 𝑥𝐵 = 𝐶))
107, 9imbi12d 336 . 2 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)))
11 csbhypf.3 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
126, 10, 11chvar 2379 1 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wnfc 2926  csb 3726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-sbc 3632  df-csb 3727
This theorem is referenced by:  disji2  4825  disjprg  4837  disjxun  4839  tfisi  7290  coe1fzgsumdlem  19990  evl1gsumdlem  20039  iundisj2  23654  disji2f  29899  disjif2  29903  iundisj2f  29912  iundisj2fi  30066
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