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

Step	Hyp	Ref	Expression
1		hbsbd.2	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nfi 1397	. . 3 ⊢ Ⅎ𝑧𝜑
3		hbsbd.3	. . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))
4	1, 3	nfdh 1463	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓)
5	2, 4	nfim1 1509	. . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓)
6	5	nfsb 1871	. . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓)
7		hbsbd.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑)
8	7	sbrim 1879	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
9	8	nfbii 1408	. . . 4 ⊢ (Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓) ↔ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓))
10	6, 9	mpbi 144	. . 3 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜓)
11	2, 10	nfrimi 1464	. 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
12	11	nfrd 1459	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))

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)