Theorem sbcocom 1893

Description: Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

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

Proof of Theorem sbcocom

Step	Hyp	Ref	Expression
1		equsb1 1716	. . 3 ⊢ [𝑧 / 𝑦]𝑦 = 𝑧
2		sbequ 1769	. . . 4 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	2	sbimi 1695	. . 3 ⊢ ([𝑧 / 𝑦]𝑦 = 𝑧 → [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 3	ax-mp 7	. 2 ⊢ [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
5		sbbi 1882	. 2 ⊢ ([𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑))
6	4, 5	mpbi 144	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]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-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:  sbcomv  1894  sbco3xzyz  1896  sbcom  1898