Theorem sbcom2v2 1903

Description: Lemma for proving sbcom2 1904. It is the same as sbcom2v 1902 but removes the distinct variable constraint on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2v2	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑤,𝑧   𝑦,𝑧

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

Proof of Theorem sbcom2v2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v 1902	. . 3 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑤 / 𝑧][𝑣 / 𝑥]𝜑)
2		sbcom2v 1902	. . . 4 ⊢ ([𝑤 / 𝑧][𝑣 / 𝑥]𝜑 ↔ [𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
3	2	sbbii 1688	. . 3 ⊢ ([𝑦 / 𝑣][𝑤 / 𝑧][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
4	1, 3	bitri 182	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
5		ax-17 1459	. . . 4 ⊢ (𝜑 → ∀𝑣𝜑)
6	5	sbco2v 1862	. . 3 ⊢ ([𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
7	6	sbbii 1688	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
8		ax-17 1459	. . 3 ⊢ ([𝑤 / 𝑧]𝜑 → ∀𝑣[𝑤 / 𝑧]𝜑)
9	8	sbco2v 1862	. 2 ⊢ ([𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
10	4, 7, 9	3bitr3i 208	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   ↔ wb 103  [wsb 1685

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbcom2  1904