Theorem hbsbd 2033

Description: Deduction version of hbsb 2000. (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

Step	Hyp	Ref	Expression
1		hbsbd.2	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nfi 1508	. . 3 ⊢ Ⅎ𝑧𝜑
3		hbsbd.3	. . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))
4	1, 3	nfdh 1570	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓)
5	2, 4	nfim1 1617	. . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓)
6	5	nfsb 1997	. . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓)
7		hbsbd.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑)
8	7	sbrim 2007	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
9	8	nfbii 1519	. . . 4 ⊢ (Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓))
10	6, 9	mpbi 145	. . 3 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓)
11	2, 10	nfrimi 1571	. 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
12	11	nfrd 1566	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1393  Ⅎwnf 1506  [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by: (None)