Theorem sbco4lem 2035

Description: Lemma for sbco4 2036. 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 2016	. . 3 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑)
2	1	sbbii 1789	. 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 1552	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
4	3	sbco2 1994	. . . . . 6 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
5	4	sbbii 1789	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
6	5	sbbii 1789	. . . 4 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
7	6	sbbii 1789	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
8		nfv 1552	. . . 4 ⊢ Ⅎ𝑤[𝑦 / 𝑥][𝑣 / 𝑦]𝜑
9	8	sbco2 1994	. . 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 1552	. . . . 5 ⊢ Ⅎ𝑣[𝑤 / 𝑦]𝜑
12	11	sbid2 1874	. . . 4 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑)
13	12	sbbii 1789	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
14	13	sbbii 1789	. 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 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbco4  2036