Theorem nfsbt 2029

Description: Closed form of nfsb 1999. (Contributed by Jim Kingdon, 9-May-2018.)

Assertion

Ref	Expression
nfsbt	⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbt

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1574	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → ∀𝑤∀𝑥Ⅎ𝑧𝜑)
2		nfsbxyt 1996	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
3	2	alimi 1503	. . . 4 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → ∀𝑤Ⅎ𝑧[𝑤 / 𝑥]𝜑)
4		nfsbxyt 1996	. . . 4 ⊢ (∀𝑤Ⅎ𝑧[𝑤 / 𝑥]𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5	3, 4	syl 14	. . 3 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
6		nfv 1576	. . . . 5 ⊢ Ⅎ𝑤𝜑
7	6	sbco2 2018	. . . 4 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	7	nfbii 1521	. . 3 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
9	5, 8	sylib 122	. 2 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
10	1, 9	syl 14	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1395  Ⅎwnf 1508  [wsb 1810

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  nfsbd  2030  setindft  16560