Theorem sbco2v 1863

Description: This is a version of sbco2 1881 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)

Hypothesis

Ref	Expression
sbco2v.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
sbco2v	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem sbco2v

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2v.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sbco2vlem 1862	. . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
3	2	sbbii 1689	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
4		ax-17 1460	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
5	4	sbco2vlem 1862	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
6		ax-17 1460	. . 3 ⊢ (𝜑 → ∀𝑤𝜑)
7	6	sbco2vlem 1862	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	3, 5, 7	3bitr3i 208	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1283  [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  nfsb  1864  equsb3  1867  sbn  1868  sbim  1869  sbor  1870  sban  1871  sbco2vd  1883  sbco3v  1885  sbcom2v2  1904  sbcom2  1905  dfsb7  1909  sb7f  1910  sbal  1918  sbal1  1920  sbex  1922