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Theorem hbsbd 1907
 Description: Deduction version of hbsb 1872. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Hypotheses
Ref Expression
hbsbd.1 (𝜑 → ∀𝑥𝜑)
hbsbd.2 (𝜑 → ∀𝑧𝜑)
hbsbd.3 (𝜑 → (𝜓 → ∀𝑧𝜓))
Assertion
Ref Expression
hbsbd (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem hbsbd
StepHypRef Expression
1 hbsbd.2 . . . 4 (𝜑 → ∀𝑧𝜑)
21nfi 1397 . . 3 𝑧𝜑
3 hbsbd.3 . . . . . . 7 (𝜑 → (𝜓 → ∀𝑧𝜓))
41, 3nfdh 1463 . . . . . 6 (𝜑 → Ⅎ𝑧𝜓)
52, 4nfim1 1509 . . . . 5 𝑧(𝜑𝜓)
65nfsb 1871 . . . 4 𝑧[𝑦 / 𝑥](𝜑𝜓)
7 hbsbd.1 . . . . . 6 (𝜑 → ∀𝑥𝜑)
87sbrim 1879 . . . . 5 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
98nfbii 1408 . . . 4 (Ⅎ𝑧[𝑦 / 𝑥](𝜑𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓))
106, 9mpbi 144 . . 3 𝑧(𝜑 → [𝑦 / 𝑥]𝜓)
112, 10nfrimi 1464 . 2 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
1211nfrd 1459 1 (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1288  Ⅎwnf 1395  [wsb 1693 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474 This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694 This theorem is referenced by: (None)
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