Theorem nfsbt 1850

Description: Closed form of nfsb 1822. (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 1419	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → ∀𝑤∀𝑥Ⅎ𝑧𝜑)
2		nfsbxyt 1819	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
3	2	alimi 1344	. . . 4 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → ∀𝑤Ⅎ𝑧[𝑤 / 𝑥]𝜑)
4		nfsbxyt 1819	. . . 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 1421	. . . . 5 ⊢ Ⅎ𝑤𝜑
7	6	sbco2 1839	. . . 4 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	7	nfbii 1362	. . 3 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
9	5, 8	sylib 127	. 2 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
10	1, 9	syl 14	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  nfsbd  1851  setindft  10090