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Theorem csbhypf 3922
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3538 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 2920 . . 3 𝑥 𝑦 = 𝐴
3 nfcsb1v 3918 . . . 4 𝑥𝑦 / 𝑥𝐵
4 csbhypf.2 . . . 4 𝑥𝐶
53, 4nfeq 2916 . . 3 𝑥𝑦 / 𝑥𝐵 = 𝐶
62, 5nfim 1899 . 2 𝑥(𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
7 eqeq1 2736 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
8 csbeq1a 3907 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
98eqeq1d 2734 . . 3 (𝑥 = 𝑦 → (𝐵 = 𝐶𝑦 / 𝑥𝐵 = 𝐶))
107, 9imbi12d 344 . 2 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)))
11 csbhypf.3 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
126, 10, 11chvarfv 2233 1 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wnfc 2883  csb 3893
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-sbc 3778  df-csb 3894
This theorem is referenced by:  disji2  5130  disjprgw  5143  disjprg  5144  disjxun  5146  tfisi  7847  coe1fzgsumdlem  21824  evl1gsumdlem  21874  iundisj2  25065  disji2f  31803  disjif2  31807  iundisj2f  31816  iundisj2fi  32003
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