Theorem hbsbd 2042

Description: Deduction version of hbsb 2009. (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 1515	. . 3 ⊢ Ⅎ𝑧𝜑
3		hbsbd.3	. . . . . . 7 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))
4	1, 3	nfdh 1577	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓)
5	2, 4	nfim1 1624	. . . . 5 ⊢ Ⅎ𝑧(𝜑 → 𝜓)
6	5	nfsb 2006	. . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑥](𝜑 → 𝜓)
7		hbsbd.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑)
8	7	sbrim 2016	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
9	8	nfbii 1526	. . . 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 1578	. 2 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
12	11	nfrd 1573	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∀𝑧[𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1400  Ⅎwnf 1513  [wsb 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by: (None)