Theorem sbco3 1897

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Assertion

Ref	Expression
sbco3	⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Proof of Theorem sbco3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco3xzyz 1896	. . 3 ⊢ ([𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
2	1	sbbii 1696	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
3		ax-17 1465	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑤[𝑦 / 𝑥]𝜑)
4	3	sbco2h 1887	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
5		ax-17 1465	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 → ∀𝑤[𝑥 / 𝑦]𝜑)
6	5	sbco2h 1887	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
7	2, 4, 6	3bitr3i 209	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by:  sbcom  1898