Theorem sbco 2021

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 1834	. . 3 ⊢ [𝑦 / 𝑥]𝑦 = 𝑥
2		sbequ12 1819	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
3	2	bicomd 141	. . . 4 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
4	3	sbimi 1812	. . 3 ⊢ ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑))
5	1, 4	ax-mp 5	. 2 ⊢ [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑)
6		sbbi 2012	. 2 ⊢ ([𝑦 / 𝑥]([𝑥 / 𝑦]𝜑 ↔ 𝜑) ↔ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	5, 6	mpbi 145	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by:  sbco3v  2022