Theorem sbco4lem 1999

Description: Lemma for sbco4 2000. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.)

Assertion

Ref	Expression
sbco4lem	⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Distinct variable groups:   𝑤,𝑣,𝜑   𝑥,𝑣,𝑤   𝑦,𝑣,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbco4lem

Step	Hyp	Ref	Expression
1		sbcom2 1980	. . 3 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑)
2	1	sbbii 1758	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑)
3		nfv 1521	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	3	sbco2 1958	. . . . . 6 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
5	4	sbbii 1758	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
6	5	sbbii 1758	. . . 4 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
7	6	sbbii 1758	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
8		nfv 1521	. . . 4 ⊢ Ⅎ𝑤[𝑦 / 𝑥][𝑣 / 𝑦]𝜑
9	8	sbco2 1958	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
10	7, 9	bitri 183	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
11		nfv 1521	. . . . 5 ⊢ Ⅎ𝑣[𝑤 / 𝑦]𝜑
12	11	sbid2 1843	. . . 4 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑)
13	12	sbbii 1758	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
14	13	sbbii 1758	. 2 ⊢ ([𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
15	2, 10, 14	3bitr3i 209	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1755

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbco4  2000