Theorem sbco2vh 1996

Description: This is a version of sbco2 2016 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)

Hypothesis

Ref	Expression
sbco2vh.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
sbco2vh	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem sbco2vh

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2vh.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sbco2vlem 1995	. . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
3	2	sbbii 1811	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
4		ax-17 1572	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
5	4	sbco2vlem 1995	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
6		ax-17 1572	. . 3 ⊢ (𝜑 → ∀𝑤𝜑)
7	6	sbco2vlem 1995	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	3, 5, 7	3bitr3i 210	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393  [wsb 1808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  nfsb  1997  equsb3  2002  sbn  2003  sbim  2004  sbor  2005  sban  2006  sbco2vd  2018  sbco3v  2020  sbcom2v2  2037  sbcom2  2038  dfsb7  2042  sb7f  2043  sbal  2051  sbal1  2053  sbex  2055