Theorem sbco3xzyz 1985

Description: Version of sbco3 1986 with distinct variable constraints between 𝑥 and 𝑧, and 𝑦 and 𝑧. Lemma for proving sbco3 1986. (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 1984	. 2 ⊢ ([𝑧 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
2		sbcocom 1982	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
3		sbcocom 1982	. 2 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4i 212	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbco3  1986