Theorem nfsbt 1947

Description: Closed form of nfsb 1917. (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 1506	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → ∀𝑤∀𝑥Ⅎ𝑧𝜑)
2		nfsbxyt 1914	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
3	2	alimi 1431	. . . 4 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → ∀𝑤Ⅎ𝑧[𝑤 / 𝑥]𝜑)
4		nfsbxyt 1914	. . . 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 1508	. . . . 5 ⊢ Ⅎ𝑤𝜑
7	6	sbco2 1936	. . . 4 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	7	nfbii 1449	. . 3 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
9	5, 8	sylib 121	. 2 ⊢ (∀𝑤∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
10	1, 9	syl 14	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1329  Ⅎwnf 1436  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  nfsbd  1948  setindft  13152