Theorem sbco 2510

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2375. See sbcov 2254 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 2509	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2		sbid 2253	. . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2074	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitri 275	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063

This theorem is referenced by:  sbid2  2511  sbco3  2516