Theorem sbco4lem 2006

Description: Lemma for sbco4 2007. 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 1987	. . 3 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑)
2	1	sbbii 1765	. 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 1528	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	3	sbco2 1965	. . . . . 6 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
5	4	sbbii 1765	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
6	5	sbbii 1765	. . . 4 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
7	6	sbbii 1765	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
8		nfv 1528	. . . 4 ⊢ Ⅎ𝑤[𝑦 / 𝑥][𝑣 / 𝑦]𝜑
9	8	sbco2 1965	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
10	7, 9	bitri 184	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
11		nfv 1528	. . . . 5 ⊢ Ⅎ𝑣[𝑤 / 𝑦]𝜑
12	11	sbid2 1850	. . . 4 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑)
13	12	sbbii 1765	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
14	13	sbbii 1765	. 2 ⊢ ([𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
15	2, 10, 14	3bitr3i 210	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbco4  2007