Theorem sbco3xzyz 1847

Description: Version of sbco3 1848 with distinct variable constraints between 𝑥 and 𝑧, and 𝑦 and 𝑧. Lemma for proving sbco3 1848. (Contributed by Jim Kingdon, 22-Mar-2018.)

Assertion

Ref	Expression
sbco3xzyz	⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem sbco3xzyz

Step	Hyp	Ref	Expression
1		sbcomxyyz 1846	. 2 ⊢ ([𝑧 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
2		sbcocom 1844	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
3		sbcocom 1844	. 2 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4i 201	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 98  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbco3  1848