Theorem sbco 1883

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		equsb2 1709	. . 3 ⊢ [𝑦 / 𝑥]𝑦 = 𝑥
2		sbequ12 1694	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
3	2	bicomd 139	. . . 4 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
4	3	sbimi 1687	. . 3 ⊢ ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑))
5	1, 4	ax-mp 7	. 2 ⊢ [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑)
6		sbbi 1874	. 2 ⊢ ([𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑) ↔ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	5, 6	mpbi 143	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 103  [wsb 1685

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbco3v  1884