Theorem sbcocom 1944

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 1759	. . 3 ⊢ [𝑧 / 𝑦]𝑦 = 𝑧
2		sbequ 1813	. . . 4 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	2	sbimi 1738	. . 3 ⊢ ([𝑧 / 𝑦]𝑦 = 𝑧 → [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 3	ax-mp 5	. 2 ⊢ [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
5		sbbi 1933	. 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 1736

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sbcomv  1945  sbco3xzyz  1947  sbcom  1949