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Theorem hbsbd 1962
Description: Deduction version of hbsb 1929. (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 1442 . . 3 𝑧𝜑
3 hbsbd.3 . . . . . . 7 (𝜑 → (𝜓 → ∀𝑧𝜓))
41, 3nfdh 1504 . . . . . 6 (𝜑 → Ⅎ𝑧𝜓)
52, 4nfim1 1551 . . . . 5 𝑧(𝜑𝜓)
65nfsb 1926 . . . 4 𝑧[𝑦 / 𝑥](𝜑𝜓)
7 hbsbd.1 . . . . . 6 (𝜑 → ∀𝑥𝜑)
87sbrim 1936 . . . . 5 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
98nfbii 1453 . . . 4 (Ⅎ𝑧[𝑦 / 𝑥](𝜑𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓))
106, 9mpbi 144 . . 3 𝑧(𝜑 → [𝑦 / 𝑥]𝜓)
112, 10nfrimi 1505 . 2 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
1211nfrd 1500 1 (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wnf 1440  [wsb 1742
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by: (None)
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