Theorem sbco 2501

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2366. See sbcov 2243 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		sbcom3 2500	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2		sbid 2242	. . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2071	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitri 274	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  sbid2  2502  sbco3  2507