Theorem sbco3xzyz 1953

Description: Version of sbco3 1954 with distinct variable constraints between 𝑥 and 𝑧, and 𝑦 and 𝑧. Lemma for proving sbco3 1954. (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 1952	. 2 ⊢ ([𝑧 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
2		sbcocom 1950	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
3		sbcocom 1950	. 2 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4i 211	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1742

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743

This theorem is referenced by:  sbco3  1954