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Theorem csbhypf 3087
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2779 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 2324 . . 3 𝑥 𝑦 = 𝐴
3 nfcsb1v 3082 . . . 4 𝑥𝑦 / 𝑥𝐵
4 csbhypf.2 . . . 4 𝑥𝐶
53, 4nfeq 2320 . . 3 𝑥𝑦 / 𝑥𝐵 = 𝐶
62, 5nfim 1565 . 2 𝑥(𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
7 eqeq1 2177 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
8 csbeq1a 3058 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
98eqeq1d 2179 . . 3 (𝑥 = 𝑦 → (𝐵 = 𝐶𝑦 / 𝑥𝐵 = 𝐶))
107, 9imbi12d 233 . 2 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)))
11 csbhypf.3 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
126, 10, 11chvar 1750 1 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wnfc 2299  csb 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050
This theorem is referenced by:  disji2  3980  tfisi  4569
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