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Theorem csbhypf 3920
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3529 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 2910 . . 3 𝑥 𝑦 = 𝐴
3 nfcsb1v 3916 . . . 4 𝑥𝑦 / 𝑥𝐵
4 csbhypf.2 . . . 4 𝑥𝐶
53, 4nfeq 2906 . . 3 𝑥𝑦 / 𝑥𝐵 = 𝐶
62, 5nfim 1892 . 2 𝑥(𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
7 eqeq1 2730 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
8 csbeq1a 3905 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
98eqeq1d 2728 . . 3 (𝑥 = 𝑦 → (𝐵 = 𝐶𝑦 / 𝑥𝐵 = 𝐶))
107, 9imbi12d 343 . 2 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)))
11 csbhypf.3 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
126, 10, 11chvarfv 2229 1 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wnfc 2876  csb 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-sbc 3776  df-csb 3892
This theorem is referenced by:  disji2  5127  disjprgw  5140  disjprg  5141  disjxun  5143  tfisi  7861  coe1fzgsumdlem  22291  evl1gsumdlem  22344  iundisj2  25566  disji2f  32497  disjif2  32501  iundisj2f  32510  iundisj2fi  32702  evl1gprodd  41829
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