Theorem sbcocom 1886

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 1709	. . 3 ⊢ [𝑧 / 𝑦]𝑦 = 𝑧
2		sbequ 1762	. . . 4 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	2	sbimi 1688	. . 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 1875	. 2 ⊢ ([𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑))
6	4, 5	mpbi 143	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)

Syntax hints:   ↔ wb 103  [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  sbcomv  1887  sbco3xzyz  1889  sbcom  1891